Unitary units in modular group algebras

TL;DR

This paper investigates the structure of unitary units in modular group algebras over fields of prime characteristic, characterizing when certain subgroups are normal and when bicyclic units are unitary.

Contribution

It provides new characterizations of normality of the unitary subgroup and conditions for bicyclic units to be unitary in modular group algebras.

Findings

Identifies conditions for the subgroup of unitary units to be normal in V.

Characterizes when all bicyclic units are unitary.

Provides structural insights into modular group algebras of p-groups.

Abstract

Let p be a prime, K a field of characteristic p, G a locally finite p-group, KG the group algebra, and V the group of the units of KG with augmentation 1. The anti-automorphism g\mapsto g^{-1} of G extends linearly to KG; this extension leaves V setwise invariant, and its restriction to V followed by v\mapsto v^{-1} lives an automorphism of V. The elements of V fixed by this automorphism are called unitary; they form a subgroup. Our first theorem describes the K and G for which this subgroup is normal in V. For each element g in G, let \bar{g} denote the sum (in KG) of the distinct powers of g. The elements 1+(g-1)h\bar{g} with g,h\in G are the bicyclic units of KG. Our second theorem describes the K and G for which all bicyclic units are unitary.