Unitary and symmetric units of a commutative group algebra

TL;DR

This paper investigates the structure of unitary and symmetric units in the group algebra of a finite abelian 2-group over the field of two elements, focusing on automorphisms called involutions.

Contribution

It provides explicit calculations of the orders and invariants of the unitary and symmetric subgroups under involutory automorphisms in FG.

Findings

Calculated the orders of unitary and symmetric subgroups.

Determined invariants of these subgroups.

Analyzed the impact of involutory automorphisms on FG units.

Abstract

Let F be the field of two elements and G a finite abelian 2-group with an involutory automorphism. The extension of this automorphism to the group algebra FG is called an involutory involution. This determines the groups of unitary and symmetric normalized units of FG. We calculate the orders and the invariants of these subgroups.