A dual view of the Clifford theory of characters of finite groups. II

Richard L. Roth
1973 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Introduction. This paper continues the analysis of Clifford theory for the case of a finite group G, K a normal subgroup of G and G/K abelian which was developed in [7] . In [7] the permutation actions of G/K on the characters of K and of (G/K)" on the characters of G were studied in relation to their effects on induction and restriction of group characters. With % an irreducible character of G, and a an irreducible component of X\K, the chain of subgroups was investigated, where I = 1(a) is
more » ... here I = 1(a) is the usual inertial subgroup for a and / = J(x) is a subgroup called the dual inertial group for %. Corresponding to the orbit of x under (G/K)" and of a under G/K we investigated a tableau of characters on /. In this paper a similar tableau is developed for /. A further subgroup M, called an intermediary subgroup, is introduced with J ^ M ^ I which has the property that a extends to a character p of M and p G = x-There are in fact e K (x) such choices for p forming one orbit under the actions of I/M and of (M/J)". (Here, the two types of actions are observed on the same set of characters.) The permutations involved are in fact identical, which leads to an isomorphism of I/M and (M/J)". Thus also I/M = M/J. M is not unique and an example is given with two intermediary subgroups M u M 2 with Mi/J gk M 2 /J. Since writing [7] , the author has become aware that some of the results on "dual Clifford theory" had been previously established in [4, Section 4] and [5, Section 3]; see also the more recent article [9, (Section 1)]. It should further be remarked that Dade, using a somewhat different approach, has also investigated the special properties of Clifford theory for G/K abelian in [1, Chapter 3].
doi:10.4153/cjm-1973-118-5 fatcat:f4gltaivgfbpfftv3p56lpuhoe