Identification of non-isomorphic $2$-groups with dihedral central quotient and isomorphic modular group algebras

Abstract

The question whether non-isomorphic finite $p$-groups can have isomorphic modular group algebras was recently answered in the negative by García-Lucas, Margolis and del Río [J. Reine Angew. Math. 783 (2022), 269–274]. We embed these negative solutions in the class of two-generated finite $2$-groups with dihedral central quotient, and solve the original question for all groups within this class. As a result, we discover new negative solutions and simple algebra isomorphisms.