Jekyll2019-04-09T09:44:01+00:00https://nickpgill.github.io/feed.xmlNick GillA Rodgers-Saxl type conjecture for characters2019-04-05T00:00:00+00:002019-04-05T00:00:00+00:00https://nickpgill.github.io/a-rodgers-saxl-conjecture-for-characters<script type="text/x-mathjax-config">
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<p>This is a follow-up to my <a href="https://nickpgill.github.io/a-rodgers-saxl-theorem">previous post on a generalization of a theorem of Rodgers and Saxl</a>.</p>
<p>The story starts with a beautiful talk I heard by Martin Liebeck in which he outlined a result due to him, Shalev & Tiep:</p>
<p><strong>Theorem</strong>: Let $f$ be a faithful irreducible of $Sym(n)$. Then $f^{*4n}$ (i.e.tensor product $4n$ times) contains every irreducible as a constituent.</p>
<p>This theorem can be interpreted as giving an upper bound on the diameter of the <a href="https://en.wikipedia.org/wiki/McKay_graph">McKay graphs</a> of the symmetric group. I won’t pursue this point of view, but it was within this context that the LST-team were working when they proved the theorem. I’m just stating the symmetric group version of their result – they had more general statements for finite (almost) simple groups.</p>
<p>Notice that the LPT-bound is pretty much as good as one could hope for: if $f^{*c}$ is to contain every irreducible as a constituent (for some positive integer $c$), then one needs $(\dim(f))^c > \sum \dim(f_i)$ where the sum on the right hand side ranges over all irreducibles of $Sym(n)$. Now the theory of Frobenius-Schur indicators tells us that, since all complex reps of $Sym(n)$ are defined over the reals, then $\sum \dim(f_i)$ is equal to 1+ the number of elements of order $2$. Writing $I_n$ for this latter quantity, <a href="https://projecteuclid.org/download/pdf_1/euclid.bams/1183553478">a result of Moser and Wyman</a> asserts that</p>
<script type="math/tex; mode=display">I_n \sim\frac{1}{\sqrt{2}} n^{n/2} \exp(-n/2-1/4+\sqrt{n}).</script>
<p>Now, using the fact that there exists an irreducible of dimension $n-1$, we obtain that</p>
<script type="math/tex; mode=display">c>= \log (\sum \dim(f_i))/ \log(\dim(f) = \log (I_n+1)/ \log (n-1),</script>
<p>and we conclude that $c$ must be at least linear in $n$.</p>
<p>A well-known heuristic in finite group theory says that whenever one proves statements about (ordinary) characters, there is probably a statement about conjugacy classes lurking nearby (and vice versa). This heuristic sounds very wooly, but it can be made rigorous in very many different contexts, and in very many different ways.</p>
<p>Sure enough, there is a “conjugacy class version” of the theorem above – it was proved by Liebeck and Shalev:</p>
<p><strong>Theorem</strong>: There exists a constant $d$ such that if $C$ is a conjugacy class of $G$, a finite non-abelian simple group and if
<script type="math/tex">k >= d \log|G|/ \log|C|,</script>
then $C^k = G$.</p>
<p>Here we are taking products of conjugacy classes instead of tensor products of characters. But, again, the result is of the same kind – it says that by taking a product $d$ times, then you will obtain all conjugacy classes, and $d$ is as small as one could possibly ask for, up to a multiplicative constant.</p>
<p>So, now, recall that in the previous post, I made the following conjecture:</p>
<p><strong>Conjecture</strong>: There exists a constant $c$ such that if $G$ is a finite simple group, and $S_1,\dots, S_k$ are subsets of $G$ satisfying
$\Pi_{i=1}^k|S_i|\geq|G|^c$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$.</p>
<p>A special case of this conjecture occurs when our sets $S_1,\dots, S_k$ are conjugacy classes of $G$. In this case, we obtain the following statement:</p>
<p><strong>Conjecture</strong>: There exists a constant $c$ such that if $G$ is a finite simple group, and $C_1,\dots, C_k$ are conjugacy classes of $G$ satisfying
$\Pi_{i=1}^k|S_i|\geq|G|^c$, then $G=C_1\cdot C_2\cdots C_k$.</p>
<p>I don’t know how to prove this theorem, but it’s possible that it’s not out of reach. The Rodgers–Saxl theorem that started all this off implies that the conjecture is true for the family $PSL(n,q)$ with the constant $c=12$. The theorem I proved with Pyber and Szabo implies it for groups of Lie type of bounded rank, so one is left with (some of) the classicals of unbounded rank, and the alternating groups.</p>
<p>But back to chacters. What would be the character–theoretic version of the previous two conjectures? The first has, if I recall correctly, been stated by the LST-team:</p>
<p><strong>Conjecture</strong> There exists $C>0$ such that if $\chi$ is a non-trivial character of a finite simple group $G$ and if
<script type="math/tex">c>C \log(\textrm{sum of dimensions of all irreducibles of }G)/ \log(\textrm{dimension of }\chi),</script>
then $\chi^{*c}$ contains every irreducible of $G$ as a constituent.</p>
<p>The Rodgers-Saxl analogue of this would be:</p>
<p><strong>Conjecture</strong> There exists $C>0$ such that if $\chi_1,…, \chi_t$ are non-trivial characters of a finite simple group $G$ and if</p>
<script type="math/tex; mode=display">\dim(\chi_1)*\dim(\chi_2)*...*\dim(\chi_t) > (\textrm{sum of dimensions of all irreducibles of }G)^C</script>
<p>then $\chi_{1}*\cdots *\chi_{t}$ contains every irreducible of $G$ as a constituent.</p>
<p>Let’s think about whether proving something like this might be possible just for the special case of $G=Sym(n)$ (OK, it’s not simple, but almost).</p>
<p>First let me note that LST prove their result by showing that if $f$ is a faithful character of $Sym(n)$, then $f * f$ or $f *f *f *f$ always contains $\chi_{1,n-1}$, from which the result follows (one just calculates how many tensor products of $\chi_{1,n-1}$ one needs to obtain all irreducibles as constituents).</p>
<p>My go-to man for symmetric group rep theory is Mark Wildon. I dropped him an email with the following question.</p>
<p><strong>Question</strong>: Does there exist a positive integer $N$ such that if $k> N$ and $f_1, …, f_k$ are irreducible characters of Sym(n), then the tensor product of $f_1,…, f_k$ (in that order) contains an irreducible constituent isomorphic to $\chi_{1,n-1}$?</p>
<p>Mark was immediately able to shed light. This question can be answered affirmatively. Indeed the following argument (which is Mark’s) shows that something much stronger is true:</p>
<p>Let $k$ be a field and let $G$ be a finite group. On page 45 of Alperin, <em>Local Representation Theory</em>, it’s shown that if $V$ is a faithful $k$G-module then there exists $N$ such that the $N$-fold tensor product $V \otimes … \otimes V$ contains a free submodule. Since $V \otimes F \cong F \oplus … \oplus F$ (with $\dim(V)$ summands) for any free $kG$-module $F$, it follows that any product with more factors (which may be arbitrary kG-modules) also contains a free submodule.</p>
<p>In the language of characters: if $\chi$ is the character of $V$, and $\psi$ is any other character, then any character $\chi^N \times \psi$ contains the regular character.</p>
<p><strong>Corollary</strong>: there exists $M$ such that any product of any $M$ faithful irreducible characters of $Sym(n)$ contains the regular character as a constituent.</p>
<p><strong>Proof</strong>: let $P$ be the number of faithful irreducible characters of $Sym(n)$. (So $P$ is 2 less than the number of partitions of $n$, unless $n = 4$.) For each faithful character, let $N(\chi)$ be the $N$ given by Alperin’s result, and let $N = \max_\chi N(\chi)$. Take $M = NP$. Then in any product of $M$ faithful characters, some character appears at least $N$ times, and so the product contains the regular character. QED</p>
<p>That’s a brilliant start, but it doesn’t give us any information about what $M$ can be. The same argument as I gave at the top of this post implies that $M$ must be bounded below by some linear function in $n$: so, then, it it possible that one can choose $M$ to be linear in $n$?</p>
<p>Here’s what Mark had to say on the subject:</p>
<blockquote>
<p><em>Some experiments in MAGMA suggest rather intriguingly that one can take
$M = n - 1$ for $Sym(n)$. This bound is sharp for $n = 3,\dots, 10$.</em></p>
</blockquote>
<p>Is this enough evidence to make a conjecture? Hell, yeah!</p>
<p><strong>Conjecture</strong>: Suppose that $f_1,\dots, f_{n-1}$ are faithful irreducible characters of $Sym(n)$. Then $f_1* \cdots* f_{n-1}$ contains the regular character as a constituent.</p>
<p>I think of this as a “Rodgers-Saxl type conjecture for characters”. Now, the challenge is to turn it into a theorem….</p>nickgillThis is a follow-up to my previous post on a generalization of a theorem of Rodgers and Saxl. The story starts with a beautiful talk I heard by Martin Liebeck in which he outlined a result due to him, Shalev & Tiep: Theorem: Let $f$ be a faithful irreducible of $Sym(n)$. Then $f^{*4n}$ (i.e.tensor product $4n$ times) contains every irreducible as a constituent. This theorem can be interpreted as giving an upper bound on the diameter of the McKay graphs of the symmetric group. I won’t pursue this point of view, but it was within this context that the LST-team were working when they proved the theorem. I’m just stating the symmetric group version of their result – they had more general statements for finite (almost) simple groups. Notice that the LPT-bound is pretty much as good as one could hope for: if $f^{*c}$ is to contain every irreducible as a constituent (for some positive integer $c$), then one needs $(\dim(f))^c > \sum \dim(f_i)$ where the sum on the right hand side ranges over all irreducibles of $Sym(n)$. Now the theory of Frobenius-Schur indicators tells us that, since all complex reps of $Sym(n)$ are defined over the reals, then $\sum \dim(f_i)$ is equal to 1+ the number of elements of order $2$. Writing $I_n$ for this latter quantity, a result of Moser and Wyman asserts that Now, using the fact that there exists an irreducible of dimension $n-1$, we obtain that and we conclude that $c$ must be at least linear in $n$. A well-known heuristic in finite group theory says that whenever one proves statements about (ordinary) characters, there is probably a statement about conjugacy classes lurking nearby (and vice versa). This heuristic sounds very wooly, but it can be made rigorous in very many different contexts, and in very many different ways. Sure enough, there is a “conjugacy class version” of the theorem above – it was proved by Liebeck and Shalev: Theorem: There exists a constant $d$ such that if $C$ is a conjugacy class of $G$, a finite non-abelian simple group and if then $C^k = G$. Here we are taking products of conjugacy classes instead of tensor products of characters. But, again, the result is of the same kind – it says that by taking a product $d$ times, then you will obtain all conjugacy classes, and $d$ is as small as one could possibly ask for, up to a multiplicative constant. So, now, recall that in the previous post, I made the following conjecture: Conjecture: There exists a constant $c$ such that if $G$ is a finite simple group, and $S_1,\dots, S_k$ are subsets of $G$ satisfying $\Pi_{i=1}^k|S_i|\geq|G|^c$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$. A special case of this conjecture occurs when our sets $S_1,\dots, S_k$ are conjugacy classes of $G$. In this case, we obtain the following statement: Conjecture: There exists a constant $c$ such that if $G$ is a finite simple group, and $C_1,\dots, C_k$ are conjugacy classes of $G$ satisfying $\Pi_{i=1}^k|S_i|\geq|G|^c$, then $G=C_1\cdot C_2\cdots C_k$. I don’t know how to prove this theorem, but it’s possible that it’s not out of reach. The Rodgers–Saxl theorem that started all this off implies that the conjecture is true for the family $PSL(n,q)$ with the constant $c=12$. The theorem I proved with Pyber and Szabo implies it for groups of Lie type of bounded rank, so one is left with (some of) the classicals of unbounded rank, and the alternating groups. But back to chacters. What would be the character–theoretic version of the previous two conjectures? The first has, if I recall correctly, been stated by the LST-team: Conjecture There exists $C>0$ such that if $\chi$ is a non-trivial character of a finite simple group $G$ and if then $\chi^{*c}$ contains every irreducible of $G$ as a constituent. The Rodgers-Saxl analogue of this would be: Conjecture There exists $C>0$ such that if $\chi_1,…, \chi_t$ are non-trivial characters of a finite simple group $G$ and if then $\chi_{1}*\cdots *\chi_{t}$ contains every irreducible of $G$ as a constituent. Let’s think about whether proving something like this might be possible just for the special case of $G=Sym(n)$ (OK, it’s not simple, but almost). First let me note that LST prove their result by showing that if $f$ is a faithful character of $Sym(n)$, then $f * f$ or $f *f *f *f$ always contains $\chi_{1,n-1}$, from which the result follows (one just calculates how many tensor products of $\chi_{1,n-1}$ one needs to obtain all irreducibles as constituents). My go-to man for symmetric group rep theory is Mark Wildon. I dropped him an email with the following question. Question: Does there exist a positive integer $N$ such that if $k> N$ and $f_1, …, f_k$ are irreducible characters of Sym(n), then the tensor product of $f_1,…, f_k$ (in that order) contains an irreducible constituent isomorphic to $\chi_{1,n-1}$? Mark was immediately able to shed light. This question can be answered affirmatively. Indeed the following argument (which is Mark’s) shows that something much stronger is true: Let $k$ be a field and let $G$ be a finite group. On page 45 of Alperin, Local Representation Theory, it’s shown that if $V$ is a faithful $k$G-module then there exists $N$ such that the $N$-fold tensor product $V \otimes … \otimes V$ contains a free submodule. Since $V \otimes F \cong F \oplus … \oplus F$ (with $\dim(V)$ summands) for any free $kG$-module $F$, it follows that any product with more factors (which may be arbitrary kG-modules) also contains a free submodule. In the language of characters: if $\chi$ is the character of $V$, and $\psi$ is any other character, then any character $\chi^N \times \psi$ contains the regular character. Corollary: there exists $M$ such that any product of any $M$ faithful irreducible characters of $Sym(n)$ contains the regular character as a constituent. Proof: let $P$ be the number of faithful irreducible characters of $Sym(n)$. (So $P$ is 2 less than the number of partitions of $n$, unless $n = 4$.) For each faithful character, let $N(\chi)$ be the $N$ given by Alperin’s result, and let $N = \max_\chi N(\chi)$. Take $M = NP$. Then in any product of $M$ faithful characters, some character appears at least $N$ times, and so the product contains the regular character. QED That’s a brilliant start, but it doesn’t give us any information about what $M$ can be. The same argument as I gave at the top of this post implies that $M$ must be bounded below by some linear function in $n$: so, then, it it possible that one can choose $M$ to be linear in $n$? Here’s what Mark had to say on the subject: Some experiments in MAGMA suggest rather intriguingly that one can take $M = n - 1$ for $Sym(n)$. This bound is sharp for $n = 3,\dots, 10$. Is this enough evidence to make a conjecture? Hell, yeah! Conjecture: Suppose that $f_1,\dots, f_{n-1}$ are faithful irreducible characters of $Sym(n)$. Then $f_1* \cdots* f_{n-1}$ contains the regular character as a constituent. I think of this as a “Rodgers-Saxl type conjecture for characters”. Now, the challenge is to turn it into a theorem….A Rodgers-Saxl type theorem2019-03-29T00:00:00+00:002019-03-29T00:00:00+00:00https://nickpgill.github.io/a-rodgers-saxl-theorem<script type="text/x-mathjax-config">
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<p>Pyber, Szabó and I have recently completed a paper entitled <em>A generalization of a theorem of Rodgers and Saxl for simple groups of bounded rank</em>. A copy of the paper can be found <a href="https://arxiv.org/abs/1901.09255">on the arXiv</a>.</p>
<p>Our result was inspired by a result of Rodgers and Saxl which appeared in <em>Communications of Algebra</em> in 2003:</p>
<p><strong>Theorem</strong>: Suppose that $C_1,\dots, C_k$ are conjugacy classes in $SL_n(q)$ such that
$\Pi_{i=1}^k|C_i|\geq|SL_n(q)|^{12}$. Then
$\Pi_{i=1}^kC_i=SL_n(q)$.</p>
<p>Our result has a similar flavour.</p>
<p><strong>Theorem</strong>: Let $G=G_r(q)$ be a finite simple group of Lie type of rank $r$. There exists $c=f(r)$ such that if $S_1,\dots, S_k$ are subsets of $G$ satisfying
$\Pi_{i=1}^k|S_i|\geq|G|^c$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$.</p>
<p>Our theorem differs to that of Rodgers and Saxl in three important respects, two good, one not so good: First, our result pertains to all finite simple groups $G$ of Lie type. Second, our result does not just pertain to conjugacy classes, but to subsets of the group, provided we are free to take conjugates.</p>
<p>The third difference is a weak point: our result replaces the constant ``12’’ in their thereom with an unspecified constant that depends on the rank of the group $G$. We conjecture that we should be able to do better, and not just for finite simple groups of Lie type, but for alternating groups as well:</p>
<p><strong>Conjecture</strong>: Let $G$ be a finite simple group. There exists $c$ such that if $S_1,\dots, S_k$ are subsets of $G$ satisfying
$\Pi_{i=1}^k|S_i|\geq|G|^c$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$.</p>
<p>This conjecture seems hard! Our theorem has a rank-dependency because it makes use of the “Product Theorem” which was proved, independently by Pyber-Szabó and by Breuillard-Green-Tao. To prove the conjecture we would need to replace the Product Theorem in our argument with, um, something else… But what?!</p>
<p>One last remark: there is a fourth sense in which our theorem differs to that of Rodgers and Saxl – we are interested in finite simple groups, while they consider $SL_n(q)$ which is, in general, only quasi-simple.</p>
<p>It turns out that the distinction here is not significant: It is not hard to show that our theorem is true if and only if the analogous statement is true for quasi-simple groups (provided you require that the sets $S_i$ do not intersect the centre of $G$)… And the same is true of the conjecture stated above. So the stated conjecture would be a generalization of both our result and that of Rodgers and Saxl, albeit we don’t specify the value of $c$ as Rodgers and Saxl did.</p>
<p>Let me finish this post by thanking my two co-authors, Laci and Bandi. I have worked with these two guys on a previous paper, and they are both brilliant and generous with their many ideas. I hope to have the privilege of working with them more in the future.</p>nickgillPyber, Szabó and I have recently completed a paper entitled A generalization of a theorem of Rodgers and Saxl for simple groups of bounded rank. A copy of the paper can be found on the arXiv. Our result was inspired by a result of Rodgers and Saxl which appeared in Communications of Algebra in 2003: Theorem: Suppose that $C_1,\dots, C_k$ are conjugacy classes in $SL_n(q)$ such that $\Pi_{i=1}^k|C_i|\geq|SL_n(q)|^{12}$. Then $\Pi_{i=1}^kC_i=SL_n(q)$. Our result has a similar flavour. Theorem: Let $G=G_r(q)$ be a finite simple group of Lie type of rank $r$. There exists $c=f(r)$ such that if $S_1,\dots, S_k$ are subsets of $G$ satisfying $\Pi_{i=1}^k|S_i|\geq|G|^c$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$. Our theorem differs to that of Rodgers and Saxl in three important respects, two good, one not so good: First, our result pertains to all finite simple groups $G$ of Lie type. Second, our result does not just pertain to conjugacy classes, but to subsets of the group, provided we are free to take conjugates. The third difference is a weak point: our result replaces the constant ``12’’ in their thereom with an unspecified constant that depends on the rank of the group $G$. We conjecture that we should be able to do better, and not just for finite simple groups of Lie type, but for alternating groups as well: Conjecture: Let $G$ be a finite simple group. There exists $c$ such that if $S_1,\dots, S_k$ are subsets of $G$ satisfying $\Pi_{i=1}^k|S_i|\geq|G|^c$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$. This conjecture seems hard! Our theorem has a rank-dependency because it makes use of the “Product Theorem” which was proved, independently by Pyber-Szabó and by Breuillard-Green-Tao. To prove the conjecture we would need to replace the Product Theorem in our argument with, um, something else… But what?! One last remark: there is a fourth sense in which our theorem differs to that of Rodgers and Saxl – we are interested in finite simple groups, while they consider $SL_n(q)$ which is, in general, only quasi-simple. It turns out that the distinction here is not significant: It is not hard to show that our theorem is true if and only if the analogous statement is true for quasi-simple groups (provided you require that the sets $S_i$ do not intersect the centre of $G$)… And the same is true of the conjecture stated above. So the stated conjecture would be a generalization of both our result and that of Rodgers and Saxl, albeit we don’t specify the value of $c$ as Rodgers and Saxl did. Let me finish this post by thanking my two co-authors, Laci and Bandi. I have worked with these two guys on a previous paper, and they are both brilliant and generous with their many ideas. I hope to have the privilege of working with them more in the future.Matrices for classical groups2019-01-01T00:00:00+00:002019-01-01T00:00:00+00:00https://nickpgill.github.io/matrices-for-classical-groups<script type="text/x-mathjax-config">
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<p>A finite classical group is best thought of as a group of linear operators on some vector space defined over a finite field. Which means, of course, that I can take a basis for this vector space and then represent the elements of my group as matrices.</p>
<p>However, I have almost <strong>never</strong> seen anyone do this in the literature. “Taking a basis” is thought of as a rather crude thing to do when doing linear algebra – one generally tries to write general arguments that do not refer to any particular basis. However, speaking for myself, when I’m trying to work out what the hell is going on inside a finite matrix group, I often end up trying to write down individual elements as matrices… And then hiding all this when it comes to writing up the paper!</p>
<p>This means that I have never publicly written down any of these calculations, despite using many of them over and over again. So this page is designed to be a little repository for me to note down interesting observations about such calculations… And perhaps they’ll be useful for someone else some time…</p>
<h2 id="working-with-omega_4q">Working with $\Omega_4^+(q)$</h2>
<p>This family of groups has some weird behaviour, especially when $q$ is even. For instance, let us write $\mathcal{U}$ for the set of maximal totally singular subspaces in a formed space of type $O+$ and dimension $2m$. If $q$ is even, then, provided $(m,q)\neq (2,2)$, we can define $\Omega_{2m}^+(q)$ to be the group inducing odd permutations on $\mathcal{U}$…. If $(m,q)=(2,2)$, however, the group so defined is <strong>not</strong> $\Omega_4^+(2)$, and one needs to define it in a completely different way (see Kleidman and Liebeck, p.31).</p>
<h1 id="finding-some-elements-in-omega_4q">Finding some elements in $\Omega_4^+(q)$.</h1>
<p>Let $(e_1,f_1)$ and $(e_2, f_2)$ be hyperbolic pairs, and consider the basis $\mathcal{B}={e_1, e_2, f_2, f_1}$ maintaining order. Then, one can calculate directly that $O_4^+(q)$ contains elements which are written with respect to $\mathcal{B}$ in the form</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{pmatrix}
1 & a & b & ab \\ & 1 & & b \\ & & 1 & a \\ & & & 1
\end{pmatrix}; %]]></script>
<p>these elements form a subgroup $S$ of size $q^2$; indeed, for $q$ odd, they form a Sylow $p$-subgroup of $O_4^+(q)$.</p>
<p>When $q$ is odd, one can look at orders to see directly that $S$ is actually a subgroup of $\Omega_4^+(q)$. Indeed, the same is true when $q$ is even, but to see this it is easiest to observe that $S$ is normalized by the element</p>
<script type="math/tex; mode=display">% <![CDATA[
g=\begin{pmatrix}
1 & & & \\ & & 1 & \\ & 1 & & \\ & & & 1
\end{pmatrix}. %]]></script>
<p>This element clearly takes one element $U=\langle e_1, e_2\rangle \in \mathcal{U}_2$ to another element $W=\langle e_1, f_2\rangle\in \mathcal{U}_2$. What is more $U\cap W$ has codimension $1$ in both $U$ and $W$. This allows us to conclude that $g\in SO_4^+(q)\setminus \Omega_4^+(q)$ (see Kleidman and Liebeck, p. 30). Now order considerations imply that $S$ must be a subgroup of $\Omega_4^+(q)$.</p>
<h1 id="finding-all-elements-in-omega_4q">Finding all elements in $\Omega_4^+(q)$.</h1>
<p>Let $R$ be the set of elements whose transpose is in $S$. One sees immediately that both $R$ and $S$ lie in $\Omega_4^+(q)$ and hence so does $\langle R,S\rangle$.</p>
<p>If one sets $a$ to equal $0$, while $b$ ranges across $\mathbb{F}_q$, in both $R$ and $S$, then one immediately observes a copy of $SL_2(q)$ in $\Omega_4^+(q)$. The same is true setting $b$ to equal $0$. Since these two copies effectively “avoid interaction” one immediately obtains a copy of $SL_2(q)\circ SL_2(q)$ inside $\Omega_4^+(q)$. (The central product is due to the fact that both copies of $SL_2(q)$ share $-I$ as an element.)</p>
<p>Now one can observe that $\Omega_4^+(q)$ also contains the element</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{pmatrix}
& & & 1\\ & & 1 & \\ & 1 & & \\ 1 & & &
\end{pmatrix} %]]></script>
<p>Now order considerations allow us to conclude that $\Omega_4^+(q)\cong (SL_2(q)\circ SL_2(q)):2$, and we have written down all elements of the group.</p>
<h2 id="sp_n-22a--omega_n2a">$Sp_{n-2}(2^a) < \Omega_n^+(2^a)$</h2>
<p>When $q$ is even, $\Omega_n^+(q)$ has a maximal subgroup – the stabilizer of a non-degenerate $1$-space – isomorphic to $Sp_{n-2}(q)$. I will write down the elements of this subgroup for $n=4$; the general case follows similarly.</p>
<p>First, we adjust our basis from before to be $\mathcal{C}={z_1, x_1, x_2, y_2}$ where $z_1=x_1+y_1$.</p>
<p>With respect to this basis, our quadratic form becomes</p>
<script type="math/tex; mode=display">Q(a,b,c,d) = ab+a^2+cd</script>
<p>and we see that $z_1$ is non-singular. Now simply observe that the following elements stabilize $z_1$:</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{pmatrix}
1 & & & c \\ & 1 & & \\ & c & 1 & c^2 \\ & & & 1
\end{pmatrix}. %]]></script>
<p>Doing the “transpose trick” like before, one immediately obtains a copy of $SL_2(q)$ in this stabilizer and (using the perfectness of $SL_2(q)$ for $q>3$ if necessary), one obtains that the stabilizer of $z_1$ in $\Omega_4^+(q)$ is isomorphic to $SL_2(q)\cong Sp_{2}(q)$.</p>
<p>One can do this more generally for larger $n$ – one simply has to exhibit the root groups of $Sp_{n-2}(q)$ inside the stabilizer of $z_1$ in $\Omega_4^+(q)$. There are two sorts of root group here: the first already ocur in $\Omega_{n-2}^+(q)$ and so are easy to write down in $\Omega_n^+(q)$; the second all take the form given above.</p>nickgillA finite classical group is best thought of as a group of linear operators on some vector space defined over a finite field. Which means, of course, that I can take a basis for this vector space and then represent the elements of my group as matrices. However, I have almost never seen anyone do this in the literature. “Taking a basis” is thought of as a rather crude thing to do when doing linear algebra – one generally tries to write general arguments that do not refer to any particular basis. However, speaking for myself, when I’m trying to work out what the hell is going on inside a finite matrix group, I often end up trying to write down individual elements as matrices… And then hiding all this when it comes to writing up the paper! This means that I have never publicly written down any of these calculations, despite using many of them over and over again. So this page is designed to be a little repository for me to note down interesting observations about such calculations… And perhaps they’ll be useful for someone else some time… Working with $\Omega_4^+(q)$ This family of groups has some weird behaviour, especially when $q$ is even. For instance, let us write $\mathcal{U}$ for the set of maximal totally singular subspaces in a formed space of type $O+$ and dimension $2m$. If $q$ is even, then, provided $(m,q)\neq (2,2)$, we can define $\Omega_{2m}^+(q)$ to be the group inducing odd permutations on $\mathcal{U}$…. If $(m,q)=(2,2)$, however, the group so defined is not $\Omega_4^+(2)$, and one needs to define it in a completely different way (see Kleidman and Liebeck, p.31). Finding some elements in $\Omega_4^+(q)$. Let $(e_1,f_1)$ and $(e_2, f_2)$ be hyperbolic pairs, and consider the basis $\mathcal{B}={e_1, e_2, f_2, f_1}$ maintaining order. Then, one can calculate directly that $O_4^+(q)$ contains elements which are written with respect to $\mathcal{B}$ in the form these elements form a subgroup $S$ of size $q^2$; indeed, for $q$ odd, they form a Sylow $p$-subgroup of $O_4^+(q)$. When $q$ is odd, one can look at orders to see directly that $S$ is actually a subgroup of $\Omega_4^+(q)$. Indeed, the same is true when $q$ is even, but to see this it is easiest to observe that $S$ is normalized by the element This element clearly takes one element $U=\langle e_1, e_2\rangle \in \mathcal{U}_2$ to another element $W=\langle e_1, f_2\rangle\in \mathcal{U}_2$. What is more $U\cap W$ has codimension $1$ in both $U$ and $W$. This allows us to conclude that $g\in SO_4^+(q)\setminus \Omega_4^+(q)$ (see Kleidman and Liebeck, p. 30). Now order considerations imply that $S$ must be a subgroup of $\Omega_4^+(q)$. Finding all elements in $\Omega_4^+(q)$. Let $R$ be the set of elements whose transpose is in $S$. One sees immediately that both $R$ and $S$ lie in $\Omega_4^+(q)$ and hence so does $\langle R,S\rangle$. If one sets $a$ to equal $0$, while $b$ ranges across $\mathbb{F}_q$, in both $R$ and $S$, then one immediately observes a copy of $SL_2(q)$ in $\Omega_4^+(q)$. The same is true setting $b$ to equal $0$. Since these two copies effectively “avoid interaction” one immediately obtains a copy of $SL_2(q)\circ SL_2(q)$ inside $\Omega_4^+(q)$. (The central product is due to the fact that both copies of $SL_2(q)$ share $-I$ as an element.) Now one can observe that $\Omega_4^+(q)$ also contains the element Now order considerations allow us to conclude that $\Omega_4^+(q)\cong (SL_2(q)\circ SL_2(q)):2$, and we have written down all elements of the group. $Sp_{n-2}(2^a) < \Omega_n^+(2^a)$ When $q$ is even, $\Omega_n^+(q)$ has a maximal subgroup – the stabilizer of a non-degenerate $1$-space – isomorphic to $Sp_{n-2}(q)$. I will write down the elements of this subgroup for $n=4$; the general case follows similarly. First, we adjust our basis from before to be $\mathcal{C}={z_1, x_1, x_2, y_2}$ where $z_1=x_1+y_1$. With respect to this basis, our quadratic form becomes and we see that $z_1$ is non-singular. Now simply observe that the following elements stabilize $z_1$: Doing the “transpose trick” like before, one immediately obtains a copy of $SL_2(q)$ in this stabilizer and (using the perfectness of $SL_2(q)$ for $q>3$ if necessary), one obtains that the stabilizer of $z_1$ in $\Omega_4^+(q)$ is isomorphic to $SL_2(q)\cong Sp_{2}(q)$. One can do this more generally for larger $n$ – one simply has to exhibit the root groups of $Sp_{n-2}(q)$ inside the stabilizer of $z_1$ in $\Omega_4^+(q)$. There are two sorts of root group here: the first already ocur in $\Omega_{n-2}^+(q)$ and so are easy to write down in $\Omega_n^+(q)$; the second all take the form given above.Groups with identical element counts2018-12-20T00:00:00+00:002018-12-20T00:00:00+00:00https://nickpgill.github.io/groups-with-identical-element-counts<script type="text/x-mathjax-config">
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<p>Some time ago I heard an interesting story about a remarkable mathematical fact. I can’t locate the source right now, so here is the story as I understand it.</p>
<p>If one consults the <em>Atlas of Finite Groups</em> and looks at the maximal subgroups of the Mathieu group $M_{23}$, one will observe that it has two rather different maximal subgroups, both of which have order 40320. The first is of form $PSL_3(4):2$; the second has form $2^4:A_7$.</p>
<p>It’s important to note how <strong>very different</strong> these two groups are, despite their shared order: in particular, they have different non-abelian composition factors.</p>
<p>John G. Thompson investigated these groups a little further and observed the following remarkable fact: they have <strong>exactly the same element count, by order</strong>. By which I mean that they have the same number of elements of order 2; the same number of elements of order 3; and so on. (How he realised this, I really don’t know – but I guess geniuses tend to realise more things than the rest of us…)</p>
<p>Thompson’s observation led him to make the following conjecture: <em>Suppose that two groups, G and H, have exactly the same element count, by order. Suppose, moreover, that G is solvable. Then H is solvable.</em></p>
<p>This conjecture can be interpreted as saying that it is, in principle, possible to recognise whether or not a group is solvable simply by knowing the number of elements in the group of each different order. Note, that I say “in principle”, as it is not proposing an algorithm for recognising solvability in this way – that is a harder question.</p>
<p>By way of comparison, it is easy to see that there is an algorithm to recognise <em>nilpotency</em> by knowing the number of elements in the group of each different order – one simply tests whether, for each prime <em>p</em> dividing the order of the group, the number of elements of order a power of <em>p</em> is equal to the highest power of <em>p</em> dividing the order of the group. If this is true for all <em>p</em>, then the group has a unique Sylow <em>p</em>-subgroup for each prime <em>p</em> dividing its order, and this property characterises nilpotency.</p>
<p>As I understand it, Thompson’s conjecture has been proved… but I’m not sure by whom. If anyone could point me to a source, I’d like to give appropriate credit! I don’t know if an algorithm for recognising solvability in this way has ever been written down. I’m also unable to point to the place where Thompson first made the observation about these subgroups of $M_{23}$ and where he first stated this conjecture; again any help with sources would be appreciated.</p>nickgillSome time ago I heard an interesting story about a remarkable mathematical fact. I can’t locate the source right now, so here is the story as I understand it. If one consults the Atlas of Finite Groups and looks at the maximal subgroups of the Mathieu group $M_{23}$, one will observe that it has two rather different maximal subgroups, both of which have order 40320. The first is of form $PSL_3(4):2$; the second has form $2^4:A_7$. It’s important to note how very different these two groups are, despite their shared order: in particular, they have different non-abelian composition factors. John G. Thompson investigated these groups a little further and observed the following remarkable fact: they have exactly the same element count, by order. By which I mean that they have the same number of elements of order 2; the same number of elements of order 3; and so on. (How he realised this, I really don’t know – but I guess geniuses tend to realise more things than the rest of us…) Thompson’s observation led him to make the following conjecture: Suppose that two groups, G and H, have exactly the same element count, by order. Suppose, moreover, that G is solvable. Then H is solvable. This conjecture can be interpreted as saying that it is, in principle, possible to recognise whether or not a group is solvable simply by knowing the number of elements in the group of each different order. Note, that I say “in principle”, as it is not proposing an algorithm for recognising solvability in this way – that is a harder question. By way of comparison, it is easy to see that there is an algorithm to recognise nilpotency by knowing the number of elements in the group of each different order – one simply tests whether, for each prime p dividing the order of the group, the number of elements of order a power of p is equal to the highest power of p dividing the order of the group. If this is true for all p, then the group has a unique Sylow p-subgroup for each prime p dividing its order, and this property characterises nilpotency. As I understand it, Thompson’s conjecture has been proved… but I’m not sure by whom. If anyone could point me to a source, I’d like to give appropriate credit! I don’t know if an algorithm for recognising solvability in this way has ever been written down. I’m also unable to point to the place where Thompson first made the observation about these subgroups of $M_{23}$ and where he first stated this conjecture; again any help with sources would be appreciated.On the subgroup lattice of a group2018-11-20T00:00:00+00:002018-11-20T00:00:00+00:00https://nickpgill.github.io/on-the-subgroup-lattice<p>A couple of months ago, I went to a very enjoyable <a href="https://groupsinflorence2.wordpress.com/">conference in Florence</a>. During the course of this conference, I found myself stood in a coffee queue next to Peter Pal Palfy, whom I had heard of but never had the privilege of meeting. I want to recount a very interesting bit of maths that he told me about while we were in that queue – although it concerns a very natural group theory question, I had never heard of it before.</p>
<p>Anyone who has done a little bit of group theory is familiar with the subgroup lattice of a group. This question concerns what can happen between a <em>second-maximal subgroup</em> of a group <em>G</em>, and the group <em>G</em> itself.</p>
<p>A <strong>second-maximal subgroup</strong>, <em>H</em>, of <em>G</em> is a group that is maximal in the lattice of subgroups of <em>G</em>, once we have removed <em>G</em> itself, along with all maximal subgroups of <em>G</em>. By definition, the only proper subgroups that contain such a group <em>H</em> are maximal subgroups of <em>G</em>. So we have this set-up:</p>
<p><img src="lattice.jpg" alt="Lattice image" /></p>
<p>So, observe, that our second-maximal subgroup <em>H</em> lies inside precisely <em>n</em> maximal subgroups of <em>G</em>. Now, here’s the question: <em>what are the possibilities for the number n in the diagram?</em></p>
<p>For instance, is it possible to find a group <em>G</em> which has a second-maximal subgroup <em>H</em> that lies in precisely 8 maximal subgroups of <em>G</em>. If the answer is “yes” to this question, then let’s say that 8 is an <strong>sml-number</strong>.</p>
<p>A preliminary observation: If <em>G</em> is elementary abelian of order <em>p^2</em> for some prime <em>p</em>, then the trivial subgroup is second-maximal, and basic linear algebra says that it lies in precisely <em>1+p</em> maximal subgroups of <em>G</em>. Thus, <em>p+1</em> is an sml-number, for every prime <em>p</em>.</p>
<p>With a bit of ingenuity, one can generalize this example to conclude that <em>q+1</em> is an sml-number, for every prime power <em>q</em>. What is more one can construct these examples so that the groups in question are solvable. Now, here’s a neat partial converse, due to Palfy and Pudlak: <em>If H is a second-maximal subgroup of a solvable group G, then H is contained in q+1 maximal subgroups for some prime power q</em>.</p>
<p>For solvable groups, then, we have a complete answer. What about non-solvable groups? Here, the situation is still unclear. A 1983 paper of Walter Feit established that both 7 and 11 are sml-number – remarkably, he does this by exhibiting two different second maximal subgroups <em>H</em> of the alternating group <em>A_31</em>, one of which lies in exactly 7 maximals (here <em>H</em> is the normalizer of a Sylow 31-subgroup in <em>PSL(5,2)</em>), the other in 11 (here <em>H</em> is the normalizer of a Sylow 31-subgroup in <em>PSL(3,5)</em>).</p>
<p>Subsequent work by Lucchini has shown that all numbers of the form <em>q+2</em> are sml-numbers for <em>q</em> a prime power. He has exhibited other infinite families of sml-numbers, although as far as I know Feit’s examples remain sporadic.</p>
<p>To my knowledge, a complete answer still remains to be proven. I find it quite remarkable that such a basic question should throw up such bizarre and sporadic behaviour, and thereby resist a complete solution…</p>nickgillA couple of months ago, I went to a very enjoyable conference in Florence. During the course of this conference, I found myself stood in a coffee queue next to Peter Pal Palfy, whom I had heard of but never had the privilege of meeting. I want to recount a very interesting bit of maths that he told me about while we were in that queue – although it concerns a very natural group theory question, I had never heard of it before. Anyone who has done a little bit of group theory is familiar with the subgroup lattice of a group. This question concerns what can happen between a second-maximal subgroup of a group G, and the group G itself. A second-maximal subgroup, H, of G is a group that is maximal in the lattice of subgroups of G, once we have removed G itself, along with all maximal subgroups of G. By definition, the only proper subgroups that contain such a group H are maximal subgroups of G. So we have this set-up: So, observe, that our second-maximal subgroup H lies inside precisely n maximal subgroups of G. Now, here’s the question: what are the possibilities for the number n in the diagram? For instance, is it possible to find a group G which has a second-maximal subgroup H that lies in precisely 8 maximal subgroups of G. If the answer is “yes” to this question, then let’s say that 8 is an sml-number. A preliminary observation: If G is elementary abelian of order p^2 for some prime p, then the trivial subgroup is second-maximal, and basic linear algebra says that it lies in precisely 1+p maximal subgroups of G. Thus, p+1 is an sml-number, for every prime p. With a bit of ingenuity, one can generalize this example to conclude that q+1 is an sml-number, for every prime power q. What is more one can construct these examples so that the groups in question are solvable. Now, here’s a neat partial converse, due to Palfy and Pudlak: If H is a second-maximal subgroup of a solvable group G, then H is contained in q+1 maximal subgroups for some prime power q. For solvable groups, then, we have a complete answer. What about non-solvable groups? Here, the situation is still unclear. A 1983 paper of Walter Feit established that both 7 and 11 are sml-number – remarkably, he does this by exhibiting two different second maximal subgroups H of the alternating group A_31, one of which lies in exactly 7 maximals (here H is the normalizer of a Sylow 31-subgroup in PSL(5,2)), the other in 11 (here H is the normalizer of a Sylow 31-subgroup in PSL(3,5)). Subsequent work by Lucchini has shown that all numbers of the form q+2 are sml-numbers for q a prime power. He has exhibited other infinite families of sml-numbers, although as far as I know Feit’s examples remain sporadic. To my knowledge, a complete answer still remains to be proven. I find it quite remarkable that such a basic question should throw up such bizarre and sporadic behaviour, and thereby resist a complete solution…MSc thesis on matrix completion2018-10-05T00:00:00+00:002018-10-05T00:00:00+00:00https://nickpgill.github.io/2018/10/05/victor-tomno-msc-thesis<p><a href="http://users.mct.open.ac.uk/is3649/index.php">Ian Short</a> and I have had the great pleasure of helping to supervise Victor Tomno, an MSc student at Moi University, Kenya. Victor has just submitted his thesis entitled <em>The weakly sign symmetric P_{p,1}^+-matrix completion problem</em>; a copy is <a href="/MSc_VictorTomno.pdf">here</a>. I want to congratulate Victor on his hard work!</p>
<p>Victor’s work looks at the problem of taking a <em>partial matrix</em> (i.e. one which does not have all entries filled), and completing it to obtain a matrix with certain prescribed properties. In the course of this research, Victor used the theory of digraphs as well as a lot of linear algebra.</p>
<p>Ian and my involvement with Victor was facilitated by the <em>Mentoring African Research Mathematicians</em> programme of the London Mathematical Society; details of that scheme are <a href="https://www.lms.ac.uk/grants/mentoring-african-research-mathematics">here</a>. Ian and I held a 2 year grant which allowed us to visit with, and host, Kenyan mathematicians, one of whom was Victor. The grant finished up in September 2018, although Ian and my connection with Kenya endures through supervision of two PhD students. We would like to thank the LMS for their financial support – this collaboration has been a very rewarding experience for all involved.</p>
<p>I should note that, in the end, Ian and I removed ourselves from the list of Victor’s official supervisors, as Moi University regulations only allowed for two supervisors on an MSc thesis. Nonetheless we are very proud to be associated with Victor’s excellent work.</p>nickgillIan Short and I have had the great pleasure of helping to supervise Victor Tomno, an MSc student at Moi University, Kenya. Victor has just submitted his thesis entitled The weakly sign symmetric P_{p,1}^+-matrix completion problem; a copy is here. I want to congratulate Victor on his hard work! Victor’s work looks at the problem of taking a partial matrix (i.e. one which does not have all entries filled), and completing it to obtain a matrix with certain prescribed properties. In the course of this research, Victor used the theory of digraphs as well as a lot of linear algebra. Ian and my involvement with Victor was facilitated by the Mentoring African Research Mathematicians programme of the London Mathematical Society; details of that scheme are here. Ian and I held a 2 year grant which allowed us to visit with, and host, Kenyan mathematicians, one of whom was Victor. The grant finished up in September 2018, although Ian and my connection with Kenya endures through supervision of two PhD students. We would like to thank the LMS for their financial support – this collaboration has been a very rewarding experience for all involved. I should note that, in the end, Ian and I removed ourselves from the list of Victor’s official supervisors, as Moi University regulations only allowed for two supervisors on an MSc thesis. Nonetheless we are very proud to be associated with Victor’s excellent work.MMath thesis on Mathieu groups2018-04-06T00:00:00+00:002018-04-06T00:00:00+00:00https://nickpgill.github.io/2018/04/06/sam-hughes-mmath-dissertation<p>My MMath student, Sam Hughes, recently submitted his dissertation entitled “<em>Representation and character theory of the small Mathieu groups</em>”; a copy is <a href="/MMath_Sam.Hughes.pdf">here</a>.</p>
<p>The dissertation studies the ordinary (complex) character theory of <em>M<sub>11</sub></em> and <em>M<sub>12</sub></em>; it includes the foundations of character theory, as well as details on how to construct <em>M<sub>11</sub></em> and <em>M<sub>12</sub></em> via the notion of “transitive extension”. I think Sam has done a beautiful job and should be congratulated!</p>
<p>We are in the process of writing up a paper including some of Sam’s results. In fact the paper comes from a slightly different point of view. Our main result is the following:</p>
<p><strong>Theorem</strong></p>
<ol>
<li>If <em>G</em> is a sharply 5-transitive subgroup of Alt(12), then the character table of <em>G</em> is given by Table ***.</li>
<li>If <em>G</em> is a sharply 4-transitive subgroup of Alt(11), then the character table of <em>G</em> is given by Table ***.</li>
</ol>
<p>The point of this theorem is that we are able to construct the character table of <em>G</em> using only the assumption about multiple-transitivity – there is no direct reference to the Mathieu groups in this paper.</p>
<p>In the course of this research, I asked a question on MathOverflow <a href="https://mathoverflow.net/questions/293859/what-did-frobenius-prove-about-m-12">here</a>. Now seems a good time to thank the contributors to that discussion, especially Frieder Ladisch, for their help!</p>nickgillMy MMath student, Sam Hughes, recently submitted his dissertation entitled “Representation and character theory of the small Mathieu groups”; a copy is here. The dissertation studies the ordinary (complex) character theory of M11 and M12; it includes the foundations of character theory, as well as details on how to construct M11 and M12 via the notion of “transitive extension”. I think Sam has done a beautiful job and should be congratulated! We are in the process of writing up a paper including some of Sam’s results. In fact the paper comes from a slightly different point of view. Our main result is the following: Theorem If G is a sharply 5-transitive subgroup of Alt(12), then the character table of G is given by Table ***. If G is a sharply 4-transitive subgroup of Alt(11), then the character table of G is given by Table ***. The point of this theorem is that we are able to construct the character table of G using only the assumption about multiple-transitivity – there is no direct reference to the Mathieu groups in this paper. In the course of this research, I asked a question on MathOverflow here. Now seems a good time to thank the contributors to that discussion, especially Frieder Ladisch, for their help!Successful grant: on Cherlins Conjecture2017-12-19T14:27:08+00:002017-12-19T14:27:08+00:00https://nickpgill.github.io/2017/12/19/successful-grant-on-cherlins-conjecture<p>I have just found that my application for EPSRC-funding has been successful. I have been awarded funding via their “standard grant” scheme. The grant will buy me out of 50% of my teaching and admin duties, thereby allowing me to focus on the problem of proving “Cherlin’s conjecture for finite primitive binary permutation groups”.</p>
<p>The <a href="/caseforsupport_cherlin.pdf">case for support</a> document from my grant application gives details of this conjecture, its importance, and the strategies that I hope to employ to work on it.</p>
<p>Excitingly, the university has agreed to fund a PhD student as part of this research. I’ll post a brief description of what the PhD will focus on below. <strong>If you are interested, please get in touch!</strong></p>
<blockquote>
<p>This programme of doctoral research is within the study of finite permutation group theory. Motivated by questions in model theory, about 20 years ago Cherlin introduced the notion of the <em>relational complexity</em> of a permutation group G; this is a positive integer which, roughly speaking, gives an indication of how easily the group G can act homogeneously on a relational structure. Cherlin’s conjecture concerns <em>binary</em> primitive permutation groups, i.e. primitive permutation groups which have relational complexity equal to 2. It is hoped that this conjecture might be proved in the next couple of years.</p>
<p>In light of this one naturally asks, next, whether we can classify groups with larger relational complexity, or whether we can calculate the relational complexity of important families of permutation groups. Calculating the relational complexity of a permutation group can be surprisingly tricky, so these sorts of questions can hide many mysteries!</p>
<p>In the process of working in this area, the student can expect to learn a great deal about the structure of finite simple groups (especially the simple classical groups) and, in particular, will study and make use of one of the most famous theorems in mathematics, the Classification of Finite Simple Groups.</p>
</blockquote>nickgillI have just found that my application for EPSRC-funding has been successful. I have been awarded funding via their “standard grant” scheme. The grant will buy me out of 50% of my teaching and admin duties, thereby allowing me to focus on the problem of proving “Cherlin’s conjecture for finite primitive binary permutation groups”. The case for support document from my grant application gives details of this conjecture, its importance, and the strategies that I hope to employ to work on it. Excitingly, the university has agreed to fund a PhD student as part of this research. I’ll post a brief description of what the PhD will focus on below. If you are interested, please get in touch! This programme of doctoral research is within the study of finite permutation group theory. Motivated by questions in model theory, about 20 years ago Cherlin introduced the notion of the relational complexity of a permutation group G; this is a positive integer which, roughly speaking, gives an indication of how easily the group G can act homogeneously on a relational structure. Cherlin’s conjecture concerns binary primitive permutation groups, i.e. primitive permutation groups which have relational complexity equal to 2. It is hoped that this conjecture might be proved in the next couple of years. In light of this one naturally asks, next, whether we can classify groups with larger relational complexity, or whether we can calculate the relational complexity of important families of permutation groups. Calculating the relational complexity of a permutation group can be surprisingly tricky, so these sorts of questions can hide many mysteries! In the process of working in this area, the student can expect to learn a great deal about the structure of finite simple groups (especially the simple classical groups) and, in particular, will study and make use of one of the most famous theorems in mathematics, the Classification of Finite Simple Groups.Crecimiento en grupos y otras estructuras2016-07-19T08:57:38+00:002016-07-19T08:57:38+00:00https://nickpgill.github.io/2016/07/19/crecimiento-en-grupos-y-otras-estructuras<p>Esta página contiene material para las escuelas de verano a la Universidad de Costa Rica (25 – 29 julio 2016) y a la Universidad National Autónoma de Nicaragua — Leon (1 – 5 agosto 2016) sobre <strong>crecimiento en grupos y otras estructuras</strong>.</p>
<p>Los niveles de los dos cursos seran un poco differentes, pero mucho de la material sera similar.</p>
<h2 id="lecturas">Lecturas</h2>
<p>Las notas son <a href="/files/2016/07/crecimiento_all-1.pdf">aquí</a>. Los subjetos son como sigue:</p>
<ul>
<li>introducción al crecimiento;</li>
<li>acciones de grupos;</li>
<li>grafos de Cayley;</li>
<li>una conjetura de Babai.</li>
</ul>
<h2 id="referencias-sobre-grupos">Referencias sobre grupos</h2>
<p>Esta material es más clasica, entonces hay muchas referencias posibles. Si no ha estado la teoría de grupos antes, recomiendo el libro de Fraleigh.</p>
<ul>
<li>Burness, <a href="http://seis.bristol.ac.uk/~tb13602/docs/permgroups_14.pdf">Topics in permutation group theory</a>, 2014.</li>
<li>Dixon and Mortimer, <em>Permutation groups,</em> 1996.</li>
<li>Fraleigh, <a href="https://cesarperezsite.files.wordpress.com/2015/04/algebra-abstracta-primer-curso-john-b-fralec3adgh-freelibros-org.pdf">Un primer curso en álgebra abstracta</a>.</li>
<li>Gill, <a href="/2014/03/11/topics-in-group-theory/">Topics in group theory</a>. (Las primeras cinco lecturas tienen material conectada a las acciones de grupos)</li>
<li>Wielandt, <em>Finite permutation groups,</em> 1964<em>.</em></ul></li>
</ul>
<h2 id="referencias-sobre-crecimiento">Referencias sobre crecimiento</h2>
<p>La mayoría de estas referencias estan un poco avanzadas. Yo he incluido dos referencias generales (por Tao y por Tao–Vu) que contienen mucho material fondamental — malafortunadamente, el libro de Tao y Vu no es disponible en una forma gratuita en la web.</p>
<p>Mi primera recomendación es las lecturas de Helfgott, “_Crecimiento y espansión en SL2″. _Primeramente, son en español(!) pero también comenzan a un nivel bastante fácil y, rapidamente, presentan un resultado muy importante de Helfgott sí mismo, sobre crecimiento en el grupo SL(2,p).</p>
<ul>
<li>Breuillard, <a href="/files/2016/07/Breuillard-Lectures-on-approximate-groups.pdf">Lectures on approximate groups</a>.</li>
<li>Green, <a href="/files/2016/07/Green-Notes-on-approximate-structures.pdf">Notes on approximate structures</a>.</li>
<li>Green, <a href="/files/2016/07/Green-Notes-on-Freimans-polynomial-conjecture.pdf">Notes on Freiman’s polynomial conjecture</a>.</li>
<li>Helfgott, <a href="/files/2016/07/Helfgott-Growth-in-groups.pdf">Growth in groups</a>.</li>
<li>Helfgott, <a href="/files/2016/07/Helfgott-Crecimiento-y-expansion-en-SL2.pdf">Crecimiento y expansión en SL2</a>.</li>
<li>Ruzsa, <a href="h/files/2016/07/Ruzsa-Sumsets-and-structure.pdf">Sumsets and structure</a>.</li>
<li>Tao, <a href="https://www.math.cmu.edu/~af1p/Teaching/AdditiveCombinatorics/Tao.pdf">Lecture notes on Additive Combinatorics</a>.
<ul>
<li>Tao, T. y Vu, V. <em>Additive Combinatorics</em>, CUP, 2006.</li>
</ul>
</li>
</ul>nickgillEsta página contiene material para las escuelas de verano a la Universidad de Costa Rica (25 – 29 julio 2016) y a la Universidad National Autónoma de Nicaragua — Leon (1 – 5 agosto 2016) sobre crecimiento en grupos y otras estructuras. Los niveles de los dos cursos seran un poco differentes, pero mucho de la material sera similar. Lecturas Las notas son aquí. Los subjetos son como sigue: introducción al crecimiento; acciones de grupos; grafos de Cayley; una conjetura de Babai. Referencias sobre grupos Esta material es más clasica, entonces hay muchas referencias posibles. Si no ha estado la teoría de grupos antes, recomiendo el libro de Fraleigh. Burness, Topics in permutation group theory, 2014. Dixon and Mortimer, Permutation groups, 1996. Fraleigh, Un primer curso en álgebra abstracta. Gill, Topics in group theory. (Las primeras cinco lecturas tienen material conectada a las acciones de grupos) Wielandt, Finite permutation groups, 1964.</ul> Referencias sobre crecimiento La mayoría de estas referencias estan un poco avanzadas. Yo he incluido dos referencias generales (por Tao y por Tao–Vu) que contienen mucho material fondamental — malafortunadamente, el libro de Tao y Vu no es disponible en una forma gratuita en la web. Mi primera recomendación es las lecturas de Helfgott, “_Crecimiento y espansión en SL2″. _Primeramente, son en español(!) pero también comenzan a un nivel bastante fácil y, rapidamente, presentan un resultado muy importante de Helfgott sí mismo, sobre crecimiento en el grupo SL(2,p). Breuillard, Lectures on approximate groups. Green, Notes on approximate structures. Green, Notes on Freiman’s polynomial conjecture. Helfgott, Growth in groups. Helfgott, Crecimiento y expansión en SL2. Ruzsa, Sumsets and structure. Tao, Lecture notes on Additive Combinatorics. Tao, T. y Vu, V. Additive Combinatorics, CUP, 2006.Successful grant: on the Product Decomposition Conjecture2015-09-23T14:27:08+00:002015-09-23T14:27:08+00:00https://nickpgill.github.io/2015/09/23/successful-grant-on-the-product-decomposition-conjecture<p>I have just found that my application for EPSRC-funding has been successful. I have been awarded funding via their “first grant” scheme. The grant will buy me out of 50% of my teaching and admin duties, thereby allowing me to focus on the problem of proving “The Product Decomposition Conjecture”.</p>
<p>The <a href="http://boolesrings.org/nickgill/files/2015/09/case-for-support.pdf">case for support</a> document from my grant application gives details of this conjecture, its importance, and the strategies that I hope to employ to work on it.</p>
<p>Excitingly, the university has agreed to fund a PhD student as part of this research. I just drafted a short description of what the PhD would be about, and I’ll post this below. (Note that this description might be edited a little over the next few days. In any case, it should give an idea of what the project will be about.) <strong>If you are interested, please get in touch!</strong></p>
<blockquote>
<p>This programme of research is within the study of finite group theory (although some investigation of linear algebraic groups may also be involved). The aim is to prove, or partially prove, the Product Decomposition Conjecture which concerns “conjugate-growth” of subsets of a finite simple group: roughly speaking, given a finite nonabelian simple group G and a subset A in G of size at least 2, we would like to show that one can always write G as a product of “not many” conjugates of A.</p>
<p>This notion of conjugate-growth has interesting connections to many interesting areas of mathematics, including expander graphs, the product growth results of Helfgott et al, bases of permutation groups, word problems and more.</p>
<p>In the process of working on this conjecture, the student can expect to learn a great deal about the structure of finite simple groups (especially the simple classical groups) and, in particular, will study and make use of one of the most famous theorems in mathematics, the Classification of Finite Simple Groups.</p>
</blockquote>nickgillI have just found that my application for EPSRC-funding has been successful. I have been awarded funding via their “first grant” scheme. The grant will buy me out of 50% of my teaching and admin duties, thereby allowing me to focus on the problem of proving “The Product Decomposition Conjecture”. The case for support document from my grant application gives details of this conjecture, its importance, and the strategies that I hope to employ to work on it. Excitingly, the university has agreed to fund a PhD student as part of this research. I just drafted a short description of what the PhD would be about, and I’ll post this below. (Note that this description might be edited a little over the next few days. In any case, it should give an idea of what the project will be about.) If you are interested, please get in touch! This programme of research is within the study of finite group theory (although some investigation of linear algebraic groups may also be involved). The aim is to prove, or partially prove, the Product Decomposition Conjecture which concerns “conjugate-growth” of subsets of a finite simple group: roughly speaking, given a finite nonabelian simple group G and a subset A in G of size at least 2, we would like to show that one can always write G as a product of “not many” conjugates of A. This notion of conjugate-growth has interesting connections to many interesting areas of mathematics, including expander graphs, the product growth results of Helfgott et al, bases of permutation groups, word problems and more. In the process of working on this conjecture, the student can expect to learn a great deal about the structure of finite simple groups (especially the simple classical groups) and, in particular, will study and make use of one of the most famous theorems in mathematics, the Classification of Finite Simple Groups.