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Character degrees and a class of finite permutation groups

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 “highfidelity” groups, By means of the regular wreath product construction it is shown that the class of highfidelity groups is "large" in the sense that every group is isomorphic to both a subgroup and a factor group of some highfidelity group. Use is made of some of D.S. Passman's results classifying soluble halftransitive groups of automorphisms in describing the structure of a special class of highfidelity 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 halftransitive 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 nontrivial elements of a faithful irreducible Gmodule 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 highfidelity group with a unique minimal normal subgroup.
