On soluble groups in which commutators have prime power order

TL;DR

This paper investigates finite soluble groups where all commutators have prime power order, revealing that the commutator subgroup's order is divisible by at most two primes, thus providing structural insights into such groups.

Contribution

It establishes a new restriction on the structure of soluble CPPO-groups, specifically limiting the prime divisors of their commutator subgroup's order.

Findings

The order of the commutator subgroup is divisible by at most two primes.

Soluble CPPO-groups have a constrained prime divisor structure.

The results deepen understanding of the structure of groups with prime power order commutators.

Abstract

The article deals with finite groups in which commutators have prime power order (CPPO-groups). We show that if G is a soluble CPPO-group, then the order of the commutator subgroup G' is divisible by at most two primes.