R. Guralnick

activity

19942026

most citedSimple groups admit Beauville structures

99 citations · 492 across the 111 of their papers we have counts for

collaborators

Showing math.GRShow all

87 papers · 1 filter

math.GR2026

Infinitely many finite simple groups of Lie type are not generated invariably by two elements of prime order

Robert M. Guralnick, John Shareshian, Russ Woodroofe

We show that there are infinitely many k for which Sp_{2k}(2) is not generated invariably by two elements of prime order.

math.GR2026

Forcible groups and Frattini covers

Sean Eberhard, Robert M. Guralnick, Carl Schildkraut +1

We say that a finite group $G$ *forces* a finite group $H$ if every finite cover of $G$ contains a subgroup isomorphic to $H$ , and we say $H$ is *forcible* if some finite group $G$ …

math.GR2026

On a conjecture of Peter Neumann on fixed points in permutation groups

Daniele Garzoni, Robert M. Guralnick, Martin W. Liebeck

We prove a conjecture of Peter Neumann from 1966, predicting that every finite non-regular primitive permutation group of degree $n$ contains an element fixing at least one point a…

math.GR2025

Derangements in finite classical groups and characteristic polynomials of random matrices

Jason Fulman, Robert Guralnick

We first obtain explicit upper bounds for the proportion of elements in a finite classical group G with a given characteristic polynomial. We use this to complete the proof that th…

math.GR2025

Commuting probability for conjugate subgroups of a finite group

Eloisa Detomi, Robert M. Guralnick, Marta Morigi +1

Given two subgroups H,K of a finite group G, the probability that a pair of random elements from H and K commutes is denoted by \pr(H,K). We address the following question. Let P b…

math.GR2024

Fixed point ratios, Sylow numbers and coverings of $p$ -elements in finite groups

Robert M. Guralnick, Attila Maróti, Juan Martínez Madrid +2

Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a p…

math.GR2024

Commuting Involutions in Finite Simple Groups

Robert M. Guralnick, Geoffrey R. Robinson

We prove that if $G$ is a finite simple group and $x, y \in G$ are involutions, then $∣ x^{G} \cap C_{G} (y) ∣ \to \infty$ as $∣ G ∣ \to \infty$ . This extends results of Gur…

math.GR2024

An extension of Gow's theorem

Robert M. Guralnick, Pham Huu Tiep

We extend Gow's theorem on products of semisimple regular conjugacy classes to finite groups whose generalized Fitting subgroup is Z(G)S where S is a quasisimple group of Lie type…

math.GR2024

Rank type conditions on commutators in finite groups

Cristina Acciarri, Robert M. Guralnick, Evgeny Khukhro +1

For a subgroup $S$ of a group $G$ , let $I_{G} (S)$ denote the set of commutators $[g, s] = g^{- 1} g^{s}$ , where $g \in G$ and $s \in S$ , so that $[G, S]$ is the subgroup generated by $I_{G} (S)$ .…

math.GR2024

Probabilistic Generation of Finite Almost Simple Groups

Jason Fulman, Daniele Garzoni, Robert M. Guralnick

We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability t…

math.GR2024

Factorizations of Almost Simple Groups with Applications

Robert M. Guralnick, Cai Heng Li, Lei Wang +1

The classification of factorizations $G = H K$ of finite almost simple groups, proposed by Wielandt in 1979 and pursued through several partial classifications, has remained open in o…

math.GR2024★ 1 cited

Weakly subnormal subgroups and variations of the Baer-Suzuki theorem

Robert M. Guralnick, Hung P. Tong-Viet, Gareth Tracey

A subgroup $R$ of a finite group $G$ is weakly subnormal in $G$ if $R$ is not subnormal in $G$ but it is subnormal in every proper overgroup of $R$ in $G$ . In this paper, we first…

math.GR2023★ 1 cited

Invariable generation of finite simple groups and rational homology of coset posets

Robert M. Guralnick, John Shareshian, Russ Woodroofe

We show that every finite simple group is generated invariably by a Sylow subgroup and a cyclic group. It follows that that the order complex of the coset poset of an arbitrary fin…

math.GR2023

On the maximal overgroups of Sylow subgroups of finite groups

Barbara Baumeister, Timothy C. Burness, Robert M. Guralnick +1

In this paper, we determine the finite groups with a Sylow $r$ -subgroup contained in a unique maximal subgroup. The proof involves a reduction to almost simple groups, and our main…

math.GR2022

Strongly dense free subgroups of semisimple algebraic groups II

Emmanuel Breuillard, Robert Guralnick, Michael Larsen

It was shown in Part I that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose…

math.GR2022

Strongly base-two groups

Timothy C. Burness, Robert M. Guralnick

Let $G$ be a finite group, let $H$ be a core-free subgroup and let $b (G, H)$ denote the base size for the action of $G$ on $G / H$ . Let $α (G)$ be the number of conjugacy classes of co…

math.GR2022

On the generation of simple groups by Sylow subgroups

Timothy C. Burness, Robert M. Guralnick

Let $G$ be a finite simple group of Lie type and let $P$ be a Sylow $2$ -subgroup of $G$ . In this paper, we prove that for any nontrivial element $x \in G$ , there exists $g \in G$ s…

math.GR2022★ 2 cited

On invariable generation of alternating groups by elements of prime and prime power order

Robert M. Guralnick, John Shareshian, Russ Woodroofe

We verify that every alternating group of degree at most one quadrillion is invariably generated by an element of prime order together with an element of prime power order.

math.GR2021★ 12 cited

On the commuting probability of p-elements in a finite group

Timothy C. Burness, Robert M. Guralnick, Alexander Moretó +1

Let $G$ be a finite group, let $p$ be a prime and let $Pr_{p} (G)$ be the probability that two random $p$ -elements of $G$ commute. In this paper we prove that ${\rm Pr}_p(G) > (…

math.GR2021★ 1 cited

Fixed point ratios for finite primitive groups and applications

Timothy C. Burness, Robert M. Guralnick

Let $G$ be a finite primitive permutation group on a set $Ω$ and recall that the fixed point ratio of an element $x \in G$ , denoted $fpr (x)$ , is the proportion of points in $…

math.GR2021

Topological generation of simple algebraic groups

Timothy C. Burness, Spencer Gerhardt, Robert M. Guralnick

Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^{r}$ with $r ⩾ 2$ . In this paper, we consider the genera…

math.GR2021

Coprime automorphisms of finite groups

Cristina Acciarri, Robert M. Guralnick, Pavel Shumyatsky

Let $G$ be a finite group admitting a coprime automorphism $α$ of order $e$ . Denote by $I_{G} (α)$ the set of commutators $g^{- 1} g^{α}$ , where $g \in G$ , and by $[G, α]$ the subgroup gene…

math.GR2021★ 1 cited

Finite groups can be generated by a pi-subgroup and a pi'-subgroup

Thomas Breuer, Robert M. Guralnick

Answering a question of Dan Haran and generalizing some results of Aschbacher-Guralnick and Suzuki, we prove that given a set of primes pi, any finite group can be generated by a p…

math.GR2020

Commuting involutions and elementary abelian subgroups of simple groups

Robert M. Guralnick, Geoffrey R. Robinson

Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that intersects ev…

Infinitely many finite simple groups of Lie type are not generated invariably by two elements of prime order

Forcible groups and Frattini covers

On a conjecture of Peter Neumann on fixed points in permutation groups

Derangements in finite classical groups and characteristic polynomials of random matrices

Commuting probability for conjugate subgroups of a finite group

Fixed point ratios, Sylow numbers and coverings of p-elements in finite groups

Commuting Involutions in Finite Simple Groups

An extension of Gow's theorem

Rank type conditions on commutators in finite groups

Probabilistic Generation of Finite Almost Simple Groups

Factorizations of Almost Simple Groups with Applications

Weakly subnormal subgroups and variations of the Baer-Suzuki theorem

Invariable generation of finite simple groups and rational homology of coset posets

On the maximal overgroups of Sylow subgroups of finite groups

Strongly dense free subgroups of semisimple algebraic groups II

Strongly base-two groups

On the generation of simple groups by Sylow subgroups

On invariable generation of alternating groups by elements of prime and prime power order

On the commuting probability of p-elements in a finite group

Fixed point ratios for finite primitive groups and applications

Topological generation of simple algebraic groups

Coprime automorphisms of finite groups

Finite groups can be generated by a pi-subgroup and a pi'-subgroup

Commuting involutions and elementary abelian subgroups of simple groups

Fixed point ratios, Sylow numbers and coverings of $p$ -elements in finite groups