Detecting the index of a subgroup in the subgroup lattice

De Falco, M.; de Giovanni, F.; Musella, C.; Schmidt, R.

doi:10.1090/S0002-9939-04-07638-5

Detecting the index of a subgroup in the subgroup lattice
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by M. De Falco, F. de Giovanni, C. Musella and R. Schmidt PDF

Proc. Amer. Math. Soc. 133 (2005), 979-985 Request permission

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 $G$ is a group and $H$ is a subgroup of finite index of $G$, the index $|G:H|$ cannot be recognized in the lattice ${\mathfrak {L}}(G)$ of all subgroups of $G$, 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 $|G:H|$.

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