An approximation for the number of subgroups

TL;DR

This paper introduces an invariant for finite groups, constructed via cobordism methods, which approximates the number of subgroups, including abelian and cyclic subgroups, with explicit formulas and examples.

Contribution

It presents a new invariant for finite groups that approximates subgroup counts, along with formulas and computed values for specific group families.

Findings

Invariant approximates subgroup counts in finite groups.

Formulas are provided for calculating the invariant.

Values are computed for various group families.

Abstract

Previously the second author has constructed by cobordism methods, an invariant associated to a finite group $G$. This invariant approximates the number of subgroups of a group, giving in some cases the number of abelian and cyclic subgroups. Here we explain the formulas used to obtain this invariant and we present values for some families of groups.