A Generalization Of Clifford's Correspondence

The induction, restriction and extension of characters are three basic techniques. It is very significant to construct a one to one correspondence between the irreducible characters of a finite group and the irreducible characters of its subgroup defined by character induction about the correlation of a group and its subgroup.Let S be a subnormal subgroup of a finite group G and η be a complex irreducible character of S. Then we say that (S, η) is a subnormal character pair of G. In this paper, we mainly investigate the general problem of how to construct a bijection defined by character induction between the inertia subgroup of (S, η) and G.we regard this as the starting point and then introduce a new conception which is called conjugation closed : Let G be an arbitrary group and (S,η) be a subnormal character pair of the group G. If (H, θ) is an arbitrary subnormal character pair of G, g is an arbitrary element of G and (S, η), (S,η)~g ≤ (H,θ), we always find an element h of H such that (S,η)~h = (S,η)~g, then we say that (S,η) is conjugation closed in G.On the basis of this conception we prove that a bijection defined by character induction can be constructed between the inertia subgroup of a subnormal character pair and the group G if the character pair is conjugation closed in G:where T is the inertia subgroup of the subnormal character pair (S, η)...