The polynomial of a permutation representation

Abstract

Let ( G, D) be a permutation representation of a finite group G acting on a finite set D. The cycle index of this representation is a polynomial P(G,D;x 1,…, x m ) in several variables x 1,…, x m with rational numbers as coefficients (see [1]). The restriction, made in [1], that the representation ( G, D) is faithful, is unnecessary and we put no restriction on ( G, D) whatsoever. We replace each variable x i of the cycle index P(G, D; x 1,…, x m ) by the polynomial Σ jx j , where j runs through the divisors of i. For instance, x 1→ x 1; x 2→ x 1+2 x 2; x 12→ x 1+2 x 2+3 x 3+4 x 4+6 x 6+12 x 12; etc. The resulting polynomial q(G, D; x 1,…, x m ) still has rational numbers as coefficients and has the additional property: o Theorem 1. If all the variables x 1,…x m of the polynomial q(G, D; x 1,…,x m) are replaced by integers (=whole numbers), the value of q is also an integer . The proof of Theorem 1 is based on [2]. We then investigate the polynomial q(G, D; x 1,…, x m ) further in the case that ( G, D) is the regular representation, i.e., when D=G and G acts on G by left multiplication. We prove: Theorem 2. If (G, D) is the regular representation, Theorem 1 is equivalent to the following theorem of Frobenius: The number of solutions in G of the equation x i=1 is divisible by the greatest common divisor of i and the order of G . Because of Theorem 2, we consider Theorem 1 as the extension of the above Frobenius theorem to all permutation representations. The polynomial q(G, D; x 1,…, x m ) for arbitrary permutation representations has been further investigated in [4].