Answer to a question on $A$-groups, arisen from the study of Steinitz classes

TL;DR

This paper addresses a question about the characterization of $A'$-groups related to Steinitz classes, proving the equivalence for groups with two prime divisors and providing counterexamples for groups with three primes.

Contribution

It proves the equivalence of $A'$-groups and solvable $A$-groups for groups with two prime divisors and constructs counterexamples with three primes.

Findings

Equivalence holds for groups with two prime divisors.

Counterexamples exist for groups with three primes.

Steinitz classes form a group in the studied cases.

Abstract

In this short note we answer to a question of group theory from arXiv:0910.5080. In that paper the author describes the set of realizable Steinitz classes for so-called $A'$-groups of odd order, obtained iterating some direct and semidirect products. It is clear from the definition that $A'$-groups are solvable $A$-groups, but the author left as an open question whether the converse is true. In this note we prove the converse when only two prime numbers divide the order of the group, but we show it to be false in general, producing a family of counterexamples which are metabelian and with exactly three primes dividing the order. Steinitz classes which are realizable for such groups in the family are computed and verified to form a group.