Embedding numbers for finite groups

Abstract

This note is concerned with the following problem. Let H denote a subgroup of a finite group G and let L denote a linear or one dimensional representation (i.e., a character) of H. We assume throughout that the field F is algebraically closed and is either of characteristic 0 or of prime characteristic which does not divide the order of any groups under consideration. Let GIL denote the corresponding induced representation of G. How many distinct (i.e., nonequivalent) irreducible representations appear in the decomposition of GI L into irreducible parts? (This number is just the central intertwining number of GI L, which is denoted by Ct(GI L). Cf. [1].) More specifically, we are interested in determining an upper bound on the number of distinct irreducible representations which will appear, purely in terms of the way H is embedded in G, and in terms which do not depend on the particular linear representation L of H. Two such bounds come quickly to mind. The number of classes (of conjugates) of the super group G, which we denote { G: e}, is clearly an upper bound. Dimension considerations also give [G: H] as an upper bound. We now introduce a new group theoretic invariant which heuristically is a measure of the manner in which the classes of G are distributed among the H-cosets of G. DEFINITION. Let H be a (not necessarily normal) subgroup of a finite group G. For each normal subset N of G, let +1(N) denote the number of classes (of conjugates) of G contained in N. Let +2(N) denote the number of right H-cosets of G which have nonzero intersection with N. Let +(N) = { G: e} -45(N) +42(N). We then define the embedding number of H in G, denoted by (G: H), to be the minimum of the +(N), as N is taken over all normal subsets of G. We remark that a definition of 42 using left cosets would yield the same value for (G: H) since N-' intersects the same number of left cosets as N does right cosets. Taking N= {e } where e is the identity element of the group we have (G: H) {G: e}. Taking N=G we have (G: H)? [G: H]. If H$ G, it is easy to verify that (G: H) > 1. If His a proper normal subgroup, then, taking N=H we have (G:H)<{G:e}. In the case where H is a normal subgroup of G, another number associated with the embedding of H in G is the number of classes in the factor group G/H. We call this the class number of H in G and denote it by { G: H}.