Some group theoretical mass formulae

TL;DR

This paper introduces new group-theoretical mass formulas involving automorphism groups, covering groups, and subclasses of finite groups, and explores their connections to classical identities like Rogers-Ramanujan.

Contribution

It presents novel mass formulas generalizing previous results, including those involving isoclinism classes and holomorphs, expanding the theoretical framework of group automorphism sums.

Findings

New mass formulas involving covering groups and subclasses of finite groups

Generalizations of P. Hall's formula to isoclinism classes and holomorphs

Connections established between mass formulas and Rogers-Ramanujan identities

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.