Enumeration of groups in some special varieties of $A$-groups

TL;DR

This paper establishes upper bounds on the number of isomorphism classes of groups within a specific variety defined by three distinct primes and analyzes subgroup properties within the general linear group.

Contribution

It provides new bounds on the enumeration of groups in the variety $A_pA_qA_r$ and on the orders and conjugacy classes of certain maximal subgroups.

Findings

Upper bound for groups of order n in the variety $A_pA_qA_r$

Bound on orders of subgroups in the variety $A_qA_r$ within the general linear group

Bound on the number of conjugacy classes of maximal subgroups

Abstract

We find an upper bound for the number of groups of order $n$ up to isomorphism in the variety $G = A_pA_qA_r$, where $p$, $q$ and $r$ are distinct primes. We also find a bound on the orders and on the number of conjugacy classes of subgroups that are maximal amongst the subgroups of the general linear group that are also in the variety $A_qA_r$.