The mutually normalizing regular subgroups of the holomorph of a cyclic group of prime power order

TL;DR

This paper characterizes mutually normalizing regular subgroups of the holomorph of cyclic p-groups, using gamma functions to construct a graph that encapsulates their normalizing relationships.

Contribution

It introduces a novel graph-based framework to classify and analyze regular subgroups of the holomorph of cyclic p-groups based on mutual normalizing properties.

Findings

Constructed the local normalizing graph for regular subgroups

Characterized all mutually normalizing regular subgroups of Hol(G)

Provided a compact representation of subgroup relationships

Abstract

Let $G=C_{p^n}$ be a finite cyclic p-group, and let $Hol(G)$ denote its holomorph. In this work, we find and characterize the regular subgroups of $Hol(G)$ that are mutually normalizing each other in the permutation group $Sym(G)$. We represent such regular subgroups as vertices of a graph, and we connect a pair of them by an edge when they mutually normalize each other. The approach to construct this local normalizing graph relies on the theory of gamma functions, and the final result will contain all the information about the regular subgroups of $Hol(G)$ in a compact form.