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.


INTRODUCTION
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. According to Miller, commutators [x, y] were introduced by Dedekind a few years earlier.
Commutators can act as a tool in all of group theory. For example, commutators can be used to compute Schur multiplier, Schur multiplier of a pair and nonabelian tensor squares of groups.
Basic definitions and theorems: Includes some definitions and results on the derived subgroups of nonabelian groups.
Definition 1: Hungerford (1997) let G be a group and X a subset of G. Let {H i | i ∊ I} be the family of all subgroups of G which contains X. Then ∩H i is called the subgroup of G generated by the set X and is denoted by < X >.
Theorem 2: Hungerford (1997) let G be a group and X a non empty subset of G. Then the subgroup < X > generated by X consists of all finite product finite product a 1 n 1 a 2 n 2 a 3 n 3 …a t n t (a i ∈ X, n i ∈ Z). In particular for every a ∈ G, < a > = {a n | n ∈ Z}.
Definition 3: Hungerford (1997) let G is a group. The subgroup of G generated by the set {x -1 y -1 xy | x, y ∈ G} is called the derived subgroup of G and denoted by G′.
Let G be a group and let G (1) be G ′ . Then for i ≥ 1, define G (i) = G (i-1)ʹ . The notation G (i) is called the ith derived subgroup of G. This gives a sequence of subgroups of G, each normal in preceding one: G > G (1) > G (2) > ···. Actually each G (i) is a normal subgroup of G. Burnside (1911) classified all finite groups of order p 2 q and Western (1898) obtained the classification of groups of order p 3 q, where p and q are distinct primes.
The classification of all nonabelian 2-generator groups of order p 3 q is given in the following theorem.
Theorem 4: Western (1898) Let G be a nonabelian 2generator group of order p 3 q, where p and q are distinct primes and p < q. Then G is exactly one group of the following types Eq. 1-6: where, a is any primitive root of a4 ≡ 1(mod q), q ≡ 1 (mod 4): where, a is any primitive root of a8 ≡ 1 (mod q), q ≡ 1 (mod 8): where, a is any primitive root of ap ≡ 1 (mod q), q ≡ 1 (mod p): where, a is any primitive root of ap2 ≡ 1 (mod q), q ≡ 1 (mod p 2 ): where, a is any primitive root of a p3 ≡ 1 (mod q), q ≡ 1 (mod p 3 ).
Theorem 5: Rashid et al. (2010) Let G be a nonabelian 2-generator group of order p 3 q, where p and q are distinct primes and p < q. Then, G ′ ∼ = C q, finite cyclic group of order q .
In this study, we focus on the derived subgroups of nonabelian 3-generator groups of order p 3 q where p and q are distinct primes and p < q.
The classification of all nonabelian 3-generator groups of order p 3 q is given in the following theorem.
Theorem 6: Western (1898) Let G be a nonabelian 3generator group of order p 3 q, where p and q are distinct primes and p < q. Then G is exactly one group of the following types Eq. 7-21: Main Result: Theorem 7: Let G be a nonabelian 3-generator group of order p 3 q, where p and q are distinct primes and p < q. Then G ′ ∼ = C 2 , Cq, C 2q , C p , C pq , Q 8 or A 4 , where Q 8 , A 4 are quaternion and alternating groups, respectively.
Proof: By Theorem 6, G has 15 types. If G is a group of type 6.1, then G has three generators A, B and Q and relations BAB = A -1 , AQ = QA and BQ = QB. For this group we can obtain the following relations: Then by mentioned relations for all x, y ∈ G, [x, y] = 1 or A 2 . Therefore, G ′ = {1, A 2 }, that is, G ′ ∼ = C 2 . The proof of the second type is similar to the first type.
To compute the derived subgroup for a group of type 6.3, by relations AB = BA, AQ = QA, BQB = Q -1 and [Q k , B] = Q -2k , we can obtain that G ′ ∼ = C q .
For proving 6.7, we can use the method that we used in type 6.6.
To compute G ′ for a group of type 6.10, by the number of generators and relations it is an immediate consequence that G ∼ = S 4 . Therefore, G ′ ∼ = A 4 .
Let G be a group of type 6.11, then relations A p2 = B p = Q q = 1, B -1 AB = A p+1 , AQ = QA, BQ = QB show that G ′ is isomorphic to C p .
Finally, for a group of type 6.14, the relations A p2 = B p = Q q = 1, B -1 AB = A p+1 , AQ = QA, B -1 Q B= Q b show that | G ′ | = pq and by computing the commutators, G ′ is a cyclic group of order pq. □