Isomorphism problem of Unitary Subgroups of Group Algebras

TL;DR

This paper investigates whether the normalized unitary subgroup of a modular group algebra uniquely determines the underlying finite p-group, confirming this for specific classes of finite p-groups.

Contribution

It provides new results confirming the isomorphism problem for V_* in certain classes of finite p-groups, expanding understanding of group algebra structures.

Findings

Confirmed the isomorphism problem for abelian p-groups

Confirmed the isomorphism problem for 2-groups of maximal class

Confirmed the isomorphism problem for small non-abelian 2-groups

Abstract

Let V_* be the normalized unitary subgroup of the modular group algebra FG of a finite p-group G over a finite field F with the classical involution *. We investigate the isomorphism problem for the group V_*, that asks when the group V_* is determined by its group algebra FG. We confirm it for classes of finite abelian p-groups, 2-groups of maximal class and non-abelian 2-groups of order at most 16.