Space of Invariant bilinear forms under representation of a group of order 8

TL;DR

This paper investigates the structure of invariant bilinear forms under group representations of order 8 over a field of characteristic 2, including their dimensions and non-degeneracy conditions.

Contribution

It explicitly computes the dimensions of invariant bilinear form spaces for all representations of groups of order 8 over characteristic 2 fields.

Findings

Dimensions of invariant bilinear form spaces are determined for each representation.

Conditions for the existence of non-degenerate invariant bilinear forms are established.

The study provides explicit examples and classifications for groups of order 8.

Abstract

Let G be a group of order 8 and F an algebraically closed field with char= 2. In this paper we compute the number of n degree representations of G and subsequent dimensions of the corresponding spaces of invatiant bilinear forms over the field F. We explicitly discuss about the existence of non-degenerate invariant bilinear forms.