The Polynomial of a Permutation Representation*

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; Xl ,..., x,~) in several variables xl ,..., x~ 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; xl ..... x,~) by the polynomial Z j x ~ , where j runs through the divisors of i. For instance, Xl -+ xl ; x~ --* xl + 2x2 ; xl~ ~ xx + 2x2 + 3x8 + 4x4 + 6xn + 12x12 ; etc. The resulting polynomial q(G, D; xx ,..., x,,) still has rational numbers as coefficients and has the additional property: THEOREM 1. I f all the variables xl ..... x,~ o f the polynomial q(G, D; Xl ..... xm) are replaced by integers (= whole numbers), the value o f q is also an integer. The proof of Theorem 1 is based on [2]. We then investigate the polynomial q(G, D; xx ,..., x,,) 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. I f (G, D) is the regular representation, Theorem 1 is equivalent to the following theorem o f Frobenius: The number o f solutions in G o f the equation x ~ = 1 is divisible by the greatest common divisor o f i and the order o f 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; xl ,..., x,n) for arbitrary permutation representations has been further investigated in [4].