Character degrees and a class of finite permutation groups

Abstract

Let f.c.d.(G) denote the set of the degrees of the faithful irreducible complex characters of a finite group G. (Of course f.c.d.(G) may be empty). Chapter 1 is concerned mainly with the structure of those groups G satisfying the condition that |f.c.d.(G)| = 1, groups which are labelled “high-fidelity” groups, By means of the regular wreath product construction it is shown that the class of high-fidelity groups is "large" in the sense that every group is isomorphic to both a subgroup and a factor group of some high-fidelity group. Use is made of some of D.S. Passman's results classifying soluble half-transitive groups of automorphisms in describing the structure of a special class of high-fidelity groups, namely those which are soluble with a complemented unique minimal normal subgroup. The same situation minus the condition that the unique minimal normal subgroup is complemented is studied in Chapter 2. There arises naturally a generalisation of half-transitive group action in which, instead of being identical, the orbit sizes are the same up to multiplication by powers of some prime. Such an action is called "q'-halftransitive", where q is the prime concerned. The results of Chapters 3 and 4 produce a classification, similar to Passman's classification mentioned above, of the possibilities for a finite soluble group G which acts q'-halftransitively on the non-trivial elements of a faithful irreducible G-module over the field of q elements. Many of Passman's techniques are used and, apart from one infinite family of groups and a small number of exceptions in the case q = 3, the possibilities for G turn out to be just those on Passman's list. Finally, in Chapter 5, an upper bound of 6 is obtained on the nilpotent length of a soluble high-fidelity group with a unique minimal normal subgroup