On a theorem of Iwasawa

Abstract

Iwasawa [1 ] has characterized those finite groups all of whose subgroups are either abelian, Hamiltonian, or nilpotent. He has shown that if the group order is divisible by more than two distinct primes then the group itself is either abelian, Hamiltonian, or nilpotent. The case of two primes has been treated by Miller [2] if all subgroups are abelian and by Iwasawa [1 ] if all subgroups are nilpotent. In Theorem 1 a set of groups will be characterized which may be described by a property similar to that of Iwasawa: that every group contains abelian, Hamiltonian, or nilpotent p-complements for every prime dividing the order of the group. (A subgroup H of G is called a p-complement of G if the index of H in G is equal to the order of a p-Sylow subgroup of G.) We first prove a lemma which then is generalized to apply to group properties which have inheritance characteristics similar to those of commutativity. In Theorem 2 we give an equivalent form of Theorem 1 with a somewhat more direct proof.