Abstract
Let N be a normal subgroup of the finite group X and let G = X/N. It is well known that the number k(X) = |Irr(X)| of conjugacy classes (irreducible characters) of X is bounded above by the product k(N)k(G). For more precise results one has to determine, for any θ ∈ Irr(N), the number of G-conjugates of θ and the number kθ(G) of irreducible characters of X lying above θ. Clifford theory reduces the computation of kθ(G) to the situation where N = Z is central in X, and then it only depends on the Clifford obstruction μ = μG(θ). The class number kμ(G) is defined for any μ ∈ H2(G, ℂ*), and behaves well when passing to isoclinic central extensions. Suppose that μ ≠ 1. There are examples where kμ(G) = 1, in which case X is a group of central type, and examples where kμ(G) = k(G) is as large as possible. In the first case G must be solvable (Howlett–Isaacs). The latter case can happen when G is perfect but not when G is a simple or, more generally, a quasisimple group.
© de Gruyter 2010
