Some group theoretical mass formulae

Abstract

A 'mass formula' is a formula involving a sum of reciprocals of automorphism groups orders. We provide several such formulae, e.g. ones involving covering groups of finite groups. Others generalize a formula of P.Hall, repalcing the class of abelian $p$-groups by subclasses, or by isoclinism classes of non-abelian groups, also by replacing automorphism groups by holomorphs, etc. We also note relations with the Rogers-Ramanujan and related identities.