## 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.