The polynomial of a permutation representation

AbstractLet (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;x1,…,xm) in several variables x1,…,xm 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 xi of the cycle index P(G, D; x1,…,xm) by the polynomial Σjxj, where j runs through the divisors of i. For instance, x1→x1; x2→x1+2x2; x12→x1+2x2+3x3+4x4+6x6+12x12; etc. The resulting polynomial q(G, D; x1,…,xm) still has rational numbers as coefficients and has the additional property:oTheorem 1. If all the variables x1,…xm of the polynomial q(G, D; x1,…,xm) 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; x1,…,xm) 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 xi=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; x1,…,xm) for arbitrary permutation representations has been further investigated in [4]