Scott Harper

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math.GR2026

The probability of generating finite and profinite groups

Scott Harper, Martyn Quick

Famously, every finite simple group $G$ can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate $G$ …

math.GR2024

Kronecker classes, normal coverings and chief factors of groups

Marco Fusari, Scott Harper, Pablo Spiga

For a group $G$ , a subgroup $U \leq G$ and a group $Inn (G) \leq A \leq Aut (G)$ , we say that $U$ is an $A$ -covering group of $G$ if $G = ⋃_{a \in A} U^{a}$ . A th…

math.GR2024

Derangements in intransitive groups

David Ellis, Scott Harper

Let $G$ be a nontrivial permutation group of degree $n$ . If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the…

math.GR2024

Representations of extensions of simple groups

Scott Harper, Martin W. Liebeck

Feit and Tits (1978) proved that a nontrivial projective representation of minimal dimension of a finite extension of a finite nonabelian simple group $G$ factors through a project…

math.GR2023

Minimal cover groups

Peter J. Cameron, David Craven, Hamid Reza Dorbidi +2

Let $F$ be a set of finite groups. A finite group $G$ is called an \emph{ $F$ -cover} if every group in $F$ is isomorphic to a subgroup of $G$ . An $\mat…