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Diameter bounds for arbitrary finite groups and applications
Sean Eberhard, Elena Maini, Luca Sabatini +1
We prove a strong general-purpose bound for the diameter of a finite group depending only on the diameters of its composition factors and the maximal exponent of a normal abelian s…
On finite groups with soluble centralisers
Valentina Grazian, Carmine Monetta, Gareth Tracey
We classify finite groups in which the centralisers of certain non-central elements are soluble. This includes a full structural description of groups whose non-central element cen…
Groups with conjugacy classes of coprime sizes
Rachel D. Camina, Attila Maróti, Emanuele Pacifici +6
Suppose that , are elements of a finite group lying in conjugacy classes of coprime sizes. We prove that is an abelian nor…
Subgroups of symmetric groups: enumeration and asymptotic properties
Colva M. Roney-Dougal, Gareth Tracey
In this paper, we prove that the symmetric group has subgroups, settling a conjecture of Pyber from 1993. We also derive asymptotically sharp upp…
Minimal sized generating sets of permutation groups
Derek F. Holt, Gareth Tracey
We present a randomised variant of an algorithm of Lucchini and Thakkar for finding a smallest sized generating set in a finite group, which has polynomial time expected running ti…
Weakly subnormal subgroups and variations of the Baer-Suzuki theorem
Robert M. Guralnick, Hung P. Tong-Viet, Gareth Tracey
A subgroup of a finite group is weakly subnormal in if is not subnormal in but it is subnormal in every proper overgroup of in . In this paper, we first…