CHAPTER 1

Introduction

The problem of investigating finite primitive permutation groups containing a

regular subgroup goes back more than one hundred years. Burnside investigated

first the groups of prime degree, and later showed that primitive groups containing

a cycle of prime-power degree

pa

are 2-transitive, except in the case a = 1. (A minor

error in his proof was noticed and corrected independently by Peter Neumann and

Wolfgang Knapp (see [27]).) Schur generalized this to primitive groups containing a

regular cycle of any composite order. Burnside suggested that perhaps the existence

of a regular p-subgroup B forced the primitive group to be 2-transitive, except in the

case where B is elementary abelian. However, examples had already existed, due to

W. Manning, of simply primitive groups in product action, with regular subgroups

which are direct products of cyclic subgroups of equal orders (not necessarily prime

– see [44, Theorem 25.7]). Wielandt investigated the problem extensively. Section

25 of his book [44] is devoted to the problem. To mark the contribution of Burnside,

he coined the term B-group for any group B whose presence as a regular subgroup

in a primitive group G forces G to be 2-transitive. He gave a number of classes of

examples of groups which are B-groups [44, Section 25], some due to Bercov and

Nagai. Wielandt also gave the first examples of non-abelian B-groups, proving that

all dihedral groups are B-groups [44, Theorem 25.6].

With the classification of finite simple groups, one can make much further

progress. By a result of Cameron, Neumann and Teague [8], if S denotes the set

of natural numbers n for which there exists a primitive group of degree n other

than An and Sn, then S has zero density in N. Hence, for almost all integers n, all

groups of order n are B-groups. Moreover, the 2-transitive permutation groups are

known. It is natural to extend the problem and ask also for a list of all pairs (G, B)

with G a primitive permutation group on a finite set Ω and B a regular subgroup.

This is the problem we are considering here.

Our main method of analysis is to view this situation as a group factorization:

for a pair (G, B) as above and a point α in the set Ω, the transitivity of B implies

that G = BGα, and since B is regular on Ω we have in addition that B ∩ Gα = 1.

The study of such factorizations was proposed by B. H. Neumann in his 1935

paper [38]. He called a factorization G = AB, where A, B are subgroups such

that A ∩ B = 1, a general product, and viewed it as a generalization of a direct

product (without the requirement that A and B be normal). In [38], among other

things, the equivalence was pointed out between general products G = AB and

transitive actions of G with point stabiliser A and regular subgroup B. According to

Neumann [39, p. 65], general products were later called Zappa-R´ edei-Sz´ ep products

(see [41, 46]), and moreover they had already occurred in the book of de S´eguier

[10] in 1904. Independently of [38] and in the same year, G.A. Miller wrote about

group factorizations in [37]. In particular he gave several examples of general

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