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Automorphism Group of the Holomorph of a Cyclic Group

Kazuki Sato

TL;DR

The paper investigates the automorphism structure of holomorphs of cyclic groups, focusing on the case $n=2p^e$ with odd prime $p$. By leveraging the semidirect product description of $G= ext{Hol}(C_n)$, dihedral group connections, and number-theoretic lemmas, it constructs an explicit isomorphism between $\text{Aut}(G)$ and $G$, establishing $\text{Hol}(C_n) \cong \text{Aut}(\text{Hol}(C_n))$ in this setting. This result extends the landscape of known equivalences between holomorphs and their automorphism groups, especially in the even-order regime where $Z(\text{Hol}(C_n))$ is nontrivial, and complements the established case of odd $n$ where holomorphs are complete. It thereby provides a concrete structural description of Aut$(G)$ for $G=\text{Hol}(C_{2p^e})$ and contributes to understanding automorphisms of holomorphs more broadly.

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.

Automorphism Group of the Holomorph of a Cyclic Group

TL;DR

The paper investigates the automorphism structure of holomorphs of cyclic groups, focusing on the case with odd prime . By leveraging the semidirect product description of , dihedral group connections, and number-theoretic lemmas, it constructs an explicit isomorphism between and , establishing in this setting. This result extends the landscape of known equivalences between holomorphs and their automorphism groups, especially in the even-order regime where is nontrivial, and complements the established case of odd where holomorphs are complete. It thereby provides a concrete structural description of Aut for and contributes to understanding automorphisms of holomorphs more broadly.

Abstract

We show that the holomorph of a cyclic group of order is isomorphic to its own automophism group when is twice of a power of an odd prime.
Paper Structure (6 sections, 13 theorems, 11 equations)

This paper contains 6 sections, 13 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semi-direct product
  4. Dihedral Groups
  5. Elementary number theory
  6. Main theorem

Key Result

Theorem 1.1

Let $p$ be an odd prime and $e>0$ a positive integer. Assume $n=2p^e$. Let $G=\operatorname{Hol}(C_n)$ be the holomorph of $C_n$. Then there exists an isomorphism $G \cong \operatorname{Aut}(G)$.

Theorems & Definitions (24)

  • Theorem 1.1: Theorem \ref{['main']}
  • Theorem 2.1: Isaacs
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 14 more