Automorphism Group of the Holomorph of a Cyclic Group

TL;DR

This paper proves that for cyclic groups of order twice a power of an odd prime, their holomorphs are isomorphic to their automorphism groups, revealing a specific structural property.

Contribution

It establishes a new isomorphism condition between the holomorph and automorphism group of certain cyclic groups, expanding understanding of their algebraic structure.

Findings

Holomorph of cyclic group of order 2p^k is isomorphic to its automorphism group.

Identifies a specific class of cyclic groups with this isomorphism property.

Provides a characterization of automorphism groups for these cyclic groups.

Abstract

We show that the holomorph of a cyclic group of order $n$ is isomorphic to its own automophism group when $n$ is twice of a power of an odd prime.