Search

Let $\Delta$ be a finite set and $G$ be a subgroup of $\operatorname{Sym}(\Delta)$. An irredundant base for $G$ is a sequence of points of $\Delta$ yielding a strictly descending chain of pointwise stabilisers, terminating with the trivial group. Suppose that $G$ is primitive and soluble. We determine asymptotically tight bounds for the maximum length of an irredundant base for $G$. Moreover, we disprove a conjecture of Seress on the maximum length of an irredundant base constructed by the natural greedy algorithm, and prove Cameron's Greedy Conjecture for $|G|$ odd.

Let $k$ and $l$ be integers, both at least 2. A $(k,l)$-bipartite graph is an $l$-regular bipartite multigraph with coloured bipartite sets of size $k$. Define $\chi(k,l)$ and $\mu(k,l)$ to be the minimum and maximum order of automorphism groups of $(k,l)$-bipartite graphs, respectively. We determine $\chi(k,l)$ and $\mu(k,l)$ for $k\geq 8$, and analyse the generic situation when $k$ is fixed and $l$ is large. In particular, we show that almost all such graphs have automorphism groups which fix the vertices pointwise and have order far less than $\mu(k,l)$. These graphs are intimately connected with both integer doubly-stochastic matrices and uniform set partitions; we examine the uniform distribution on the set of $k\times k$ integer doubly-stochastic matrices with all line sums $l$, showing that with high probability all entries stray far from the mean. We also show that the symmetric group acting on uniform set partitions is non-synchronizing., Comment: 18 pages

The relational complexity of a subgroup $G$ of $\mathrm{Sym}(\Omega)$ is a measure of the way in which the orbits of $G$ on $\Omega^k$ for various $k$ determine the original action of $G$. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between $\mathrm{PSL}_{n}(\mathbb{F})$ and $\mathrm{PGL}_{n}(\mathbb{F})$, for an arbitrary field $\mathbb{F}$, acting on the set of $1$-dimensional subspaces of $\mathbb{F}^n$. We also bound the relational complexity of all groups lying between $\mathrm{PSL}_{n}(q)$ and $\mathrm{P}\Gamma\mathrm{L}_{n}(q)$, and generalise these results to the action on $m$-spaces for $m \ge 1$., Comment: 20 pages. Version 2: minor changes incorporating referee comments. To appear in J. Group Theory

An irredundant base of a group $G$ acting faithfully on a finite set $\Gamma$ is a sequence of points in $\Gamma$ that produces a strictly descending chain of pointwise stabiliser subgroups in $G$, terminating at the trivial subgroup. Suppose that $G$ is $\operatorname{S}_n$ or $\operatorname{A}_n$ acting primitively on $\Gamma$, and that the point stabiliser is primitive in its natural action on $n$ points. We prove that the maximum size of an irredundant base of $G$ is $O\left(\sqrt{n}\right)$, and in most cases $O\left((\log n)^2\right)$. We also show that these bounds are best possible.

A base for a permutation group $G$ acting on a set $\Omega$ is a subset $\mathcal{B}$ of $\Omega$ such that the pointwise stabiliser $G_{(\mathcal{B})}$ is trivial. Let $n$ and $r$ be positive integers with $n>2r$. The symmetric and alternating groups $\mathrm{S}_n$ and $\mathrm{A}_n$ admit natural primitive actions on the set of $r$-element subsets of $\{1,2,\dots, n\}$. Building on work of Halasi [6], we provide explicit expressions for the base sizes of all of these actions, and hence determine the base size of all primitive actions of $\mathrm{S}_n$ and $\mathrm{A}_n$., Comment: 7 pages, 1 figure

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups $H$ contain an element $s$ such that the maximal subgroups of $H$ containing $s$, but not containing the socle of $H$, are pairwise non-conjugate., Comment: 33 pages. Incorporated referee comments, including a correction to the statement of Proposition 2.21

The normaliser problem takes as input subgroups $G$ and $H$ of the symmetric group $S_n$, and asks one to compute $N_G(H)$. The fastest known algorithm for this problem is simply exponential, whilst more efficient algorithms are known for restricted classes of groups. In this paper, we will focus on groups with many orbits. We give a new algorithm for the normaliser problem for these groups that performs many orders of magnitude faster than previous implementations in GAP. We also prove that the normaliser problem for the special case $G=S_n$ is at least as hard as computing the group of monomial automorphisms of a linear code over any field of fixed prime order.

In this paper we show that if $G$ is a primitive subgroup of $S_{n}$ that is not large base, then any irredundant base for $G$ has size at most $5 \log n$. This is the first logarithmic bound on the size of an irredundant base for such groups, and is best possible up to a small constant. As a corollary, the relational complexity of $G$ is at most $5 \log n+1$, and the maximal size of a minimal base and the height are both at most $5 \log n.$ Furthermore, we deduce that a base for $G$ of size at most $5 \log n$ can be computed in polynomial time.

The normaliser problem has as input two subgroups $H$ and $K$ of the symmetric group $S_n$, and asks for a generating set for $N_K(H)$: it is not known to have a subexponential time solution. It is proved in [Roney-Dougal & Siccha, 2020] that if $H$ is primitive then the normaliser problem can be solved in quasipolynomial time. We show that for all subgroups $H$ and $K$ of $S_n$, in quasipolynomial time we can decide whether $N_{S_n}(H)$ is primitive, and if so compute $N_K(H)$. Hence we reduce the question of whether one can solve the normaliser problem in quasipolynomial time to the case where the normaliser is known not to be primitive.

Let G be a permutation group, acting on a set \Omega of size n. A subset B of \Omega is a base for G if the pointwise stabilizer G_(B) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of Sym(n) is large base if there exist integers m and r \geq 1 such that Alt(m)^r \unlhd G \leq Sym(m) \wr Sym(r), where the action of Sym(m) is on k-element subsets of {1,...,m} and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group M24 in its natural action on 24 points, or b(G) \leq \lceil \log n\rceil+1. Furthermore, we show that there are infinitely many primitive groups G that are not large base for which b(G) > log n + 1, so our bound is optimal.

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail., Comment: 13 pages

The generating graph $\Gamma(G)$ of a finite group $G$ has vertex set the non-identity elements of $G$, with two elements connected exactly when they generate $G$. A coclique in a graph is an empty induced subgraph, so a coclique in $\Gamma(G)$ is a subset of $G$ such that no pair of elements generate $G$. A coclique is maximal if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of $G$ form a coclique in $\Gamma(G)$, but this coclique need not be maximal. In this paper we determine when the intransitive maximal subgroups of $\textrm{S}_n$ and $\textrm{A}_n$ are maximal cocliques in the generating graph. In addition, we prove a conjecture of Cameron, Lucchini, and Roney-Dougal [3] in the case of $G = \textrm{A}_n$ and $\textrm{S}_n$, when n is prime and $n \neq \frac{(q^d -1)}{(q-1)}$ for all prime powers $q$ and $d \geq 2$. Namely, we show that two elements of $G$ have identical sets of neighbours in $\Gamma(G)$ if and only if they belong to exactly the same maximal subgroups.

It is undecidable in general whether a given finitely presented group is word hyperbolic. We use the concept of pregroups, introduced by Stallings, to define a new class of van Kampen diagrams, which represent groups as quotients of virtually free groups. We then present a polynomial-time procedure which analyses these diagrams, and either returns an explicit linear Dehn function for the presentation, or returns fail, together with its reasons for failure. Furthermore, if our procedure succeeds we are often able to produce in polynomial time a word problem solver for the presentation that runs in linear time. Our algorithms have been implemented, and are often many orders of magnitude faster than KBMAG, the only comparable publicly available software., Comment: To appear in Journal of Symbolic Computation

We analyse the complexity of constructing involution centralisers in unitary groups over fields of odd order. In particular, we prove logarithmic bounds on the number of random elements required to generate a subgroup of the centraliser of a strong involution that contains the last term of its derived series. We use this to strengthen previous bounds on the complexity of recognition algorithms for unitary groups in odd characteristic. Our approach generalises and extends two previous papers by the second author and collaborators on strong involutions and regular semisimple elements of linear groups., Comment: 48 pages. Revised after suggestions from Eamonn O'Brien and an anonymous referee. To appear J. Algebra

The standard $(n, k, d)$ model of random groups is a model where the relators are chosen randomly from the set of cyclically reduced words of length $k$ on an $n$-element generating set. Gromov's density model of random groups considers the case where $n$ is fixed, and $k$ tends to infinity. We instead fix $k$, and let $n$ tend to infinity. We prove that for all $k \geq 2$ at density $d > 1/2$ a random group in this model is trivial or cyclic of order two, whilst for $d < \frac{1}{2}$ such a random group is infinite and hyperbolic. In addition we show that for $d<\frac{1}{k}$ such a random group is free, and that this threshold is sharp. These extend known results for the triangular ($k = 3$) and square ($k = 4)$ models of random groups.

We investigate the extent to which the exchange relation holds in finite groups $G$. We define a new equivalence relation $\equiv_{\mathrm{m}}$, where two elements are equivalent if each can be substituted for the other in any generating set for $G$. We then refine this to a new sequence $\equiv_{\mathrm{m}}^{(r)}$ of equivalence relations by saying that $x \equiv_{\mathrm{m}}^{(r)}y$ if each can be substituted for the other in any $r$-element generating set. The relations $\equiv_{\mathrm{m}}^{(r)}$ become finer as $r$ increases, and we define a new group invariant $\psi(G)$ to be the value of $r$ at which they stabilise to $\equiv_{\mathrm{m}}$. Remarkably, we are able to prove that if $G$ is soluble then $\psi(G) \in \{d(G), d(G) +1\}$, where $d(G)$ is the minimum number of generators of $G$, and to classify the finite soluble groups $G$ for which $\psi(G) = d(G)$. For insoluble $G$, we show that $d(G) \leq \psi(G) \leq d(G) + 5$. However, we know of no examples of groups $G$ for which $\psi(G) > d(G) + 1$. As an application, we look at the generating graph of $G$, whose vertices are the elements of $G$, the edges being the $2$-element generating sets. Our relation $\equiv_{\mathrm{m}}^{(2)}$ enables us to calculate $\mathrm{Aut}(\Gamma(G))$ for all soluble groups $G$ of nonzero spread, and give detailed structural information about $\mathrm{Aut}(\Gamma(G))$ in the insoluble case., Comment: 23 pages

We improve on recent estimates for the probability of generating the alternating and symmetric groups $\mathrm{Alt}(n)$ and $\mathrm{Sym}(n)$. In particular we find the sharp lower bound, if the probability is given by a quadratic in $n^{-1}$. This leads to improved bounds on the largest number $h(\mathrm{Alt}(n))$ such that a direct product of $h(\mathrm{Alt}(n))$ copies of $\mathrm{Alt}(n)$ can be generated by two elements.

A finite group $G$ is \emph{coprimely-invariably generated} if there exists a set of generators $\{g_1, ..., g_u\}$ of $G$ with the property that the orders $|g_1|, ..., |g_u|$ are pairwise coprime and that for all $x_1, ..., x_u \in G$ the set $\{g_1^{x_1}, ..., g_u^{x_u}\}$ generates $G$. We show that if $G$ is coprimely-invariably generated, then $G$ can be generated with three elements, or two if $G$ is soluble, and that $G$ has zero presentation rank. As a corollary, we show that if $G$ is any finite group such that no proper subgroup has the same exponent as $G$, then $G$ has zero presentation rank. Furthermore, we show that every finite simple group is coprimely-invariably generated. Along the way, we show that for each finite simple group $S$, and for each partition $\pi_1, ..., \pi_u$ of the primes dividing $|S|$, the product of the number $k_{\pi_i}(S)$ of conjugacy classes of $\pi_i$-elements satisfies $\prod_{i=1}^u k_{\pi_i}(S) \leq \frac{|S|}{2| Out S|}.$

Let Omega be a quasisimple classical group in its natural representation over a finite vector space V, and let Delta be its normaliser in the general linear group. We construct the projection from Delta to Delta/Omega and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent Delta/Omega as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Omega. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups.