arXiv subjects

Saul D. Freedman

Publications and source records attributed to Saul D. Freedman.

13 recordsLinked to original sources

arXiv2026

An infinite family of counterexamples to the Polycirculant Conjecture

We disprove the Polycirculant Conjecture, which states that every transitive 2-closed permutation group is non-elusive, i.e. contains a derangement of prime order. In fact, we prove a stronger result, answering a long-standing question of Marušič and Jordan: there exists a vertex-transitive graph admitting no semiregular automorphism. To do so, we employ recently developed methods of Chen et al. for constructing elusive groups via non-split extensions, allowing us to construct an elusive group $7^6.\mathrm{PSU}_3(3)$ of degree 16,464. We show that this group is the full automorphism group of seven of its orbital graphs and hence is 2-closed. Our example extends to infinitely many counterexamples of the Polycirculant Conjecture, and infinitely many vertex-transitive graphs admitting no semiregular automorphism.

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arXiv2026

On the generalised Saxl graphs of permutation groups

A base for a finite permutation group $G \le \mathrm{Sym}(Ω)$ is a subset of $Ω$ with trivial pointwise stabiliser in $G$, and the base size of $G$ is the smallest size of a base for $G$. Motivated by the interest in groups of base size two, Burness and Giudici introduced the notion of the Saxl graph. This graph has vertex set $Ω$, with edges between elements if they form a base for $G$. We define a generalisation of this graph that encodes useful information about $G$ whenever $b(G) \ge 2$: here, the edges are the pairs of elements of $Ω$ that can be extended to bases of size $b(G)$. In particular, for primitive groups, we investigate the completeness and arc-transitivity of the generalised graph, and the generalisation of Burness and Giudici's Common Neighbour Conjecture on the original Saxl graph.

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arXiv2024

The non-commuting, non-generating graph of a finite simple group

Let $G$ be a group such that $G/Z(G)$ is finite and simple. The non-commuting, non-generating graph $Ξ(G)$ of $G$ has vertex set $G \setminus Z(G)$, with edges corresponding to pairs of elements that do not commute and do not generate $G$. Complementing our previous investigation of this graph for non-simple groups, we show that $Ξ(G)$ is connected with diameter at most $5$, with smaller upper bounds for certain families of groups. Using these bounds, we then prove that when $G$ is simple, the diameter of the complement of the generating graph of $G$ has a tight upper bound of $4$, with the exception of at most one group with a graph of diameter $5$.

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arXiv2024

Spreading primitive groups of diagonal type do not exist

The synchronisation hierarchy of finite permutation groups consists of classes of groups lying between 2-transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabiliser is supplemented by all corresponding two-point stabilisers.

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arXiv2024

The relational complexity of linear groups acting on subspaces

The relational complexity of a subgroup $G$ of $\mathrm{Sym}(Ω)$ is a measure of the way in which the orbits of $G$ on $Ω^k$ 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 $\mathrm{P}Γ\mathrm{L}_{n}(q)$, and generalise these results to the action on $m$-spaces for $m \ge 1$.

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arXiv2023

The non-commuting, non-generating graph of a non-simple group

Let $G$ be a (finite or infinite) group such that $G/Z(G)$ is not simple. The non-commuting, non-generating graph $Ξ(G)$ of $G$ has vertex set $G \setminus Z(G)$, with vertices $x$ and $y$ adjacent whenever $[x,y] \ne 1$ and $\langle x, y \rangle \ne G$. We investigate the relationship between the structure of $G$ and the connectedness and diameter of $Ξ(G)$. In particular, we prove that the graph either: (i) is connected with diameter at most $4$; (ii) consists of isolated vertices and a connected component of diameter at most $4$; or (iii) is the union of two connected components of diameter $2$. We also describe in detail the finite groups with graphs of type (iii). In the companion paper arXiv:2212.01616, we consider the case where $G/Z(G)$ is finite and simple.

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arXiv2023

Finite groups satisfying the independence property

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.

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arXiv2023

Total closure for permutation actions of finite nonabelian simple groups

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if for each set $Ω$ upon which $G$ acts faithfully, $G$ is the largest subgroup of $\mathrm{Sym}(Ω)$ that leaves invariant each of the $G$-orbits in the induced action on $Ω\times\cdots\times Ω=Ω^k$. Each finite group $G$ is totally $|G|$-closed, and $k(G)$ denotes the least integer $k$ such that $G$ is totally $k$-closed. We address the question of determining the closure number $k(G)$ for finite simple groups $G$. Prior to our work it was known that $k(G)=2$ for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that $k(G)\geq3$ for all other finite simple groups. We determine the value for the alternating groups, namely $k(A_n)=n-1$. In addition, for all simple groups $G$, other than alternating groups and classical groups, we show that $k(G)\leq 7$. Finally, if $G$ is a finite simple classical group with natural module of dimension $n$, we show that $k(G)\leq n+2$ if $n \ge 14$, and $k(G) \le \lfloor n/3 + 12 \rfloor$ otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on $n$ and the type of $G$) on the base sizes of the primitive actions of $G$, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups.

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arXiv2022

Tournaments and Even Graphs are Equinumerous

A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwise. Pontus von Brömssen (né Andersson) showed that the existence of such an automorphism is independent of the orientation, and considered the question of counting pairwise non-isomorphic even graphs. Based on computational evidence, he made the rather surprising conjecture that the number of pairwise non-isomorphic even graphs on $n$ vertices is equal to the number of pairwise non-isomorphic tournaments on $n$ vertices. We prove this conjecture using a counting argument with several applications of the Cauchy-Frobenius Theorem.

math.CO↗

arXiv2021

The intersection graph of a finite simple group has diameter at most 5

Let $G$ be a non-abelian finite simple group. In addition, let $Δ_G$ be the intersection graph of $G$, whose vertices are the proper nontrivial subgroups of $G$, with distinct subgroups joined by an edge if and only if they intersect nontrivially. We prove that the diameter of $Δ_G$ has a tight upper bound of 5, thereby resolving a question posed by Shen (2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

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arXiv2020

The non-commuting, non-generating graph of a nilpotent group

For a nilpotent group $G$, let $Ξ(G)$ 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, $Ξ(G)$ 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 $Ξ^+(G)$ be the subgraph of $Ξ(G)$ induced by its non-isolated vertices. We show that if $Ξ(G)$ has an edge, then $Ξ^+(G)$ is connected with diameter $2$ or $3$, with $Ξ(G) = Ξ^+(G)$ 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 $Ξ(G)$ in more detail.

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arXiv2020

On $p$-groups with automorphism groups related to the exceptional Chevalley groups

Let $\hat G$ be the finite simply connected version of an exceptional Chevalley group, and let $V$ be a nontrivial irreducible module, of minimal dimension, for $\hat G$ over its field of definition. We explore the overgroup structure of $\hat G$ in $\mathrm{GL}(V)$, and the submodule structure of the exterior square (and sometimes the third Lie power) of $V$. When $\hat G$ is defined over a field of odd prime order $p$, this allows us to construct the smallest (with respect to certain properties) $p$-groups $P$ such that the group induced by $\mathrm{Aut}(P)$ on $P/Φ(P)$ is either $\hat G$ or its normaliser in $\mathrm{GL}(V)$.

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arXiv2019

On $p$-groups with automorphism groups related to the Chevalley group $G_2(p)$

Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\mathrm{Aut}(P)$ induces $N_{\mathrm{GL}(7,p)}(G_2(p)) = Z(\mathrm{GL}(7,p)) G_2(p)$ on the Frattini quotient $P/Φ(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p = 3$, our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_2(q)$ on the octonion algebra over $\mathbb{F}_q$, for each power $q$ of $p$, and explore the reducibility of the exterior square of each irreducible seven-dimensional $\mathbb{F}_q[G_2(q)]$-module.

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