Isols and burnside's lemma

Abstract

A nonnegative integer is called a number, a collection of numbers a set and a collection of sets a class. Suppose P is a partition of a finite set σ into nonempty finite sets; the members of P are called blocks. If the finite group G acts on the finite set σ, it induces a partition P_G of σ whose blocks are the orbits of G. Writing card Δ for the cardinality of a collection Δ we have

• (A) if P = (δ₁,…, δ_k), then Σ_i=1^k card δ_i = card σ,

• (B) if G = (g₁,…, g_t), β_i = {xϵσ | g_i(x) = x}, k = card P_G, then Σ_i=1^t = kt (Frobenius-Burnside).

This paper deals with generalizations of (A) and (B), in particular with the case where all blocks of P and orbits of G remain finite, while the roles of the numbers card β_i, k and t are played by isols. The main results are presented in Section 5. An application to enumeration theory is discussed in Section 6.