The modular isomorphism problem and abelian direct factors (Q6141301)

Let \(k\) be a field of positive characteristic \(p\) and \(G\) and \(H\) finite \(p\)-groups. The modular isomorphism problem asks whether the existence of an isomorphism of \(k\)-algebras \(kG\simeq kH\) implies that \(G\simeq H\). In the paper under review, the author shows that the isomorphism type of the maximal abelian direct factor of a group \(G\), as well as the isomorphism type of the group algebra over \(\mathbb{F}_{p}\) of the non-abelian remaining direct factor (if existing) are determined by \(\mathbb{F}_{p}G\), generalizing the main result by \textit{L. Margolis} et al. [J. Algebra 636, 533--559 (2023; Zbl 07754341)].