The existence of subgroups of given order in finite groups

Abstract

1. Discussion of results 1·1. Introduction: The classical theorem of Lagrange states that the order of a subgroup of a finite group G divides the order, ( G ), of G . More generally, if H and K are subgroups of G , and H ≥ K , then ( G : K ) = ( G : H )( H : K ), where ( G : K ) denotes the index of K in G , etc. We call a number a possible order of a subgroup of G if it is a divisor of ( G ), and a possible order of a subgroup of G containing a subgroup H if it is a divisor of ( G ) and a multiple of ( H ). In this paper we discuss conditions on G for the existence of subgroups of every possible order, the existence of subgroups of every possible order containing arbitrary subgroups, and similar properties.