Normal $\pi$ -complement theorems

Abstract

It is a well-known theorem of Frobenius that a finite group G has a normal p-complement if and only if two elements of its Sylow p-subgroup that are conjugate in G are already conjugate in P. This result was generalized by Brauer and Suzuki, see e.g. [2], from Sylow to Hall subgroups using additional conditions, namely if H is a Hall $\pi$ -subgroup of G and two elements of H that are conjugate in G are already conjugate in H and each elementary $\pi$ -subgroup in G can be conjugated into H, then G has a normal $\pi$ -complement. In this paper we generalize the theorem of Frobenius from Sylow to Hall subgroups under different conditions, the conjugacy condition is restricted only for elements of odd prime order and elements of order 2 and 4 in H, on the other hand we assume that H has a Sylow tower. This also generalizes a result of Zappa, see [8], saying that if H is a Hall $\pi$ -subgroup of G with a Sylow tower, and two elements of H that are conjugate in G are already conjugate in H, then G has a normal $\pi$ -complement. As a corollary we get a weakening of the conditions of another result of Zappa, saying that if a finite group has a Hall- $\pi$ -subgroup H with a Sylow tower and H possesses a set of complete right coset representatives, which is invariant under conjugation by H, then G has a normal $\pi$ -complement. In the end we generalize the theorem of Brauer and Suzuki in another direction, namely assuming that G has a solvable Hall $\pi$ -subgroup and every elementary $\pi$ -subgroup of G can be conjugated into it, and if two elements of H of prime power order in H that are conjugate in G are already conjugate in H, then G has a normal $\pi$ -complement. In this paper all groups are finite. For basic definitions the reader is referred to [6].