On the Derived Subgroups of Some Finite Groups

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.