On the Derived Subgroups of Some Finite Groups

M M

doi:10.3844/jmssp.2012.111.113

Abstract

Problem statement: In this study we focus on the derived subgroup of nonabelian 3-generator groups of order p3q, where p and q are distinct primes and p < q. Our main objective is to compute the derived subgroup for these groups up to isomorphism. Approach: In a group G, the derived subgroup G' = [G, G] is generated by the set of commutators of G, K (G) = {[x, y]| x, y ∈ G} and introduced by Dedekind. The relations of the group are used to compute the derived subgroup. Results: The results show that the derived subgroup of nonabelian 3-generator groups of order p3q is a cyclic group, Q8 or A4. Conclusion/Recommendations: The problem can be considered to compute the derived subgroup of these groups without the use of the relations.

Highlights

Miller (1898) introduced the derived subgroup G′ of a group G as the subgroup generated by K (G) = {[x, y]| x, y ∈ G}, the set of commutators of G
Where, a is any primitive root of ap2 ≡ 1, q ≡ 1: G = < A, Q | Ap2 = Qq = 1, A−1QA = Qa >
We focus on the derived subgroups of nonabelian 3-generator groups of order p3q where p and q are distinct primes and p < q

Summary

Introduction

Miller (1898) introduced the derived subgroup G′ of a group G as the subgroup generated by K (G) = {[x, y]| x, y ∈ G}, the set of commutators of G. Basic definitions and theorems: Includes some definitions and results on the derived subgroups of nonabelian groups. ∩Hi is called the subgroup of G generated by the set X and is denoted by < X >.

Results

Conclusion