(SI10-063) Number of Automorphisms of Some Non-Abelian p-Groups of Order p^4

Authors

Muniya ., Shri Jagdishprasad Jhabarmal Tibrewala University
Harsha Arora, Govt. College Hansi (Hisar)
Mahender Singh, Shri Jagdishprasad Jhabarmal Tibrewala University

Abstract

The automorphism of a group is a way of mapping the object to itself while preserving all of its structure, and the set of automorphisms of an object forms a group called the automorphism group. It is simply a bijective homomorphism. One of the earliest group automorphism was given by Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus where he discovered an order two automorphism. In this paper, we compute the automorphisms of some non-Abelian groups of order p⁴, where p is an odd prime and GAP (Group Algorithm Programming) software has been used for the verification of number of automorphisms. We use Burnside’s categorization for the classification of all non-isomorphic groups of order p⁴. There are fifteen groups of order p⁴, out of which five are Abelian and the rest are non-Abelian. The automorphisms of all five Abelian groups are already computed by researchers. We find out the automorphisms by using the structure description of group and elementary calculations. By assuming any arbitrary automorphisms, we find out the constraints for group automorphism by using these results and the property of automorphisms. This research work is good in nature and helpful in the further studies of algebra.

Recommended Citation

., Muniya; Arora, Harsha; and Singh, Mahender (2022). (SI10-063) Number of Automorphisms of Some Non-Abelian p-Groups of Order p^4, Applications and Applied Mathematics: An International Journal (AAM), Vol. 17, Iss. 3, Article 7.
Available at: https://digitalcommons.pvamu.edu/aam/vol17/iss3/7