On $G$-character tables for normal subgroups

TL;DR

This paper explores the structure of $G$-character tables for normal subgroups, providing an alternative algebraic approach and demonstrating their usefulness in deriving subgroup information.

Contribution

It offers a new perspective on the structure of $G$-character tables using group algebra, showing these matrices are non-singular and useful for subgroup analysis.

Findings

Matrices are non-singular

Character tables contain induced submatrices

Method aids in subgroup information extraction

Abstract

Let $N$ be a normal subgroup of a finite group $G$. From a result due to Brauer, it can be derived that the character table of $G$ contains square submatrices which are induced by the $G$-conjugacy classes of elements in $N$ and the $G$-orbits of irreducible characters of $N$. In the present paper, we provide an alternative approach to this fact through the structure of the group algebra. We also show that such matrices are non-singular and become a useful tool to obtain information of $N$ from the character table of $G$.