On the number of subgroups of a given exponent in a finite abelian group

TL;DR

This paper investigates the enumeration of subgroups with a specific exponent in finite abelian groups, providing explicit formulas for low ranks and an asymptotic estimate for general cases.

Contribution

It introduces explicit formulas for counting subgroups of a given exponent in rank two and three abelian groups, along with an asymptotic formula for broader cases.

Findings

Explicit formulas for rank two and three groups

Asymptotic formula for general finite abelian groups

Enhanced understanding of subgroup structure based on exponents

Abstract

This paper deals with the number of subgroups of a given exponent in a finite abelian group. Explicit formulas are obtained in the case of rank two and rank three abelian groups. An asymptotic formula is also presented.