Mathematics and Computer Science, Volume 2

20Automorphisms of Some Non-Abelian p−Groups of Order p⁴

Muniya Sansanwal^1⋆, Harsha Arora² and Mahender Singh³

¹Department of Mathematics, JJTU Jhunjhunu, Rajasthan, India

²Department of Mathematics, Govt. College for Women Hisar, Haryana, India

³Department of Mathematics, Om Sterling Global University, Hisar, Haryana, India

Abstract

This paper contains the number of automorphisms of several non-Abelian groups of order p⁴, p-odd prime are computed and GAP (Group Algorithm Programming) software has been used for the verification of a number of automorphisms.

Keywords: -p-groups, automorphism, semi-direct product

MSC subject classification 2000: 20D45, 20D60

20.1 Introduction

Let G be a p-group of order p⁴, p-odd prime and Aut(G) represent the group of all automorphisms of a given group G. There are many research papers in the literature related to the automorphisms of p-groups, for instance [1, 3, 4], etc. In [1], the automorphisms of groups of order p³ are computed along with the automorphisms of abelian groups of order p⁴. The present paper is an extension of the research work in [1]. In this paper, we will compute the automorphisms of groups of order p⁴. William Burnside [2] classified all groups having order p⁴. By using these classifications we derive the automorphisms of some groups having order p⁴. We divide the derived results into two parts. In these two parts, one part is dedicated to the categorization of all p−groups of order p⁴ and the other part is dedicated to ...