Enumeration of Groups in Varieties of $A$-groups: A survey

TL;DR

This survey reviews known results and open questions about counting finite groups within varieties of $A$-groups, highlighting differences in asymptotic behavior compared to general groups.

Contribution

It compiles and discusses existing results and open problems related to enumeration of finite $A$-groups, emphasizing their unique asymptotic properties.

Findings

Asymptotic bounds differ between abelian, $A$-groups, and general groups.

Survey highlights key results in enumeration within $A$-group varieties.

Identifies open questions in the enumeration of finite $A$-groups.

Abstract

Let $S$ be a class of groups and let $f_S (n)$ be the number of isomorphism classes of groups in $S$ of order $n$. Let $f(n)$ count the number of groups of order $n$ up to isomorphism. The asymptotic bounds for $f(n)$ behave differently when restricted to abelian groups, $A$-groups and groups in general. We survey some results and some open questions in enumeration of finite groups with a focus on enumerating within varieties of $A$-groups.