# On number of isomorphism classes of derived subgroups

## TL;DR

This paper characterizes finite nonabelian characteristically simple groups with a specific number of isomorphism classes of derived subgroups, showing that only A5 satisfies the condition n = |(G)|+2.

## Contribution

It provides a precise characterization of such groups, establishing a unique condition linking the number of derived subgroup classes to the prime divisors.

## Key findings

- A5 is the only finite nonabelian characteristically simple group with n = |(G)|+2.
- The number of isomorphism classes of derived subgroups relates directly to the prime divisors of the group.
- The paper offers a new criterion to identify A5 among similar groups.

## Abstract

In this paper we show that a finite nonabelian characteristically simple group G satisfying n = |\pi(G)|+2 if and only if G is isomorphic to A5, where n is the number of isomorphism classes of derived subgroups of G and \pi(G) is the set of prime divisors of the group G.