p-Groups in which kernels of the non-linear irreducible characters are of equal order

TL;DR

This paper characterizes finite groups of prime power order (for odd primes) where all non-linear irreducible characters have kernels of equal size, advancing understanding of group structure related to character kernels.

Contribution

It provides a complete characterization of such groups, highlighting a specific uniformity condition on the kernels of non-linear irreducible characters.

Findings

Groups of odd prime power order with uniform kernel size are classified.

The structure of these groups is explicitly described.

The results extend previous knowledge on character kernels in finite groups.

Abstract

For an irreducible character $\chi$ of a finite group $G$, its kernel is defined as $\text{ker }\chi=\{g\in G: \chi(g)=\chi(1)\}$. In this paper we characterize the finite groups of prime power order(for odd prime) in which kernels of all of the non-linear irreducible characters are of the same order.