Minimal degrees associated with some wreath products of groups

Ibrahim Alotaibi and David Easdown

Abstract

We investigate minimal degrees of groups associated with certain wreath products. We construct sequences of groups with the property that some proper quotients are isomorphic to subgroups having the same minimal degree, thus having the so-called almost exceptional  property. We show that it is possible to have an almost exceptional group with an arbitrarily long chain of normal subgroups with respect to which the quotients all have the same minimal degree, whilst at the same time having arbitrarily many subgroups, also with the same minimal degree, but which are pairwise incomparable. The results depend on a theory of semidirect products, where the base group is a \(k\)-dimensional vector space over the field with \(p\) elements, where \(p\) is a prime and \(k\) is a positive integer, extended by a cyclic group of order \(p\), represented by a \(k\times k\) matrix. This theory uncovers a large class of nonabelian groups of exponent \(p\). A final application is made to construct sequences of groups with the property that the direct products have minimal degrees that grow as a linear function of the number \(n\) of factors, whilst their respective quotients, realised as central products, have minimal degrees that grow as an exponential function of \(n\), generalising a result of Peter Neumann.

Keywords: permutation groups, wreath products, semidirect products, minimal degrees.

AMS Subject Classification: Primary 20B35.