On group invariants determined by modular group algebras: even versus odd characteristic

TL;DR

This paper investigates which structural features of finite p-groups with cyclic commutator subgroup are uniquely determined by their group algebras over fields of characteristic p, highlighting differences between odd and even primes.

Contribution

It proves that for odd primes, key invariants like the exponent and abelianization of the centralizer of the commutator subgroup are determined by the group algebra, extending to almost all invariants for 2-generated groups.

Findings

Exponent and abelianization of the centralizer are determined by the group algebra.

For 2-generated groups, most invariants are determined by the group algebra.

The isomorphism type of the centralizer of the commutator subgroup is determined for odd primes.

Abstract

Let $p$ be a an odd prime and let $G$ be a finite $p$-group with cyclic commutator subgroup $G'$. We prove that the exponent and the abelianization of the centralizer of $G'$ in $G$ are determined by the group algebra of $G$ over any field of characteristic $p$. If, additionally, $G$ is $2$-generated then almost all the numerical invariants determining $G$ up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of $G'$ is determined. These claims are known to be false for $p=2$.