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

Abstract

Let G=Cpn 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 mutually normalize each other in the symmetric group Sym(G)

. We represent regular subgroups of Hol(G) 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.

Keywords

• mutually
• normalizing
• regular
• subgroups
• holomorph
• cyclic
• group
• prime
• power
• order