Detecting the index of a subgroup in the subgroup lattice

Abstract

A theorem by Zacher and Rips states that the finiteness of the index of a subgroup can be described in terms of purely lattice-theoretic concepts. On the other hand, it is clear that if  G G is a group and H H is a subgroup of finite index of G G , the index | G : H | |G:H| cannot be recognized in the lattice L ( G ) {\mathfrak {L}}(G) of all subgroups of G G , as for instance all groups of prime order have isomorphic subgroup lattices. The aim of this paper is to give a lattice-theoretic characterization of the number of prime factors (with multiplicity) of | G : H | |G:H| .