EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 2329-2335 ISSN 1307-5543 – ejpam.com Published by New York Business Global Exploring the Associated Groups of Quasi-Free Groups Abdulaziz Mutlaq Alotaibi1, Khaled Mustafa Aljamal2,∗ 1 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia 2 Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, 21030 Kuala Nerus, Terengganu Abstract. Let G is a cyclic group. Then H(G) is a trivial group and if G = G1∗ . . . ∗Gn is the free product of the groups G1, . . . , Gn, then H(G) = H(G1 ∗ . . . ∗ G∗) ≊ H(G1) × . . . × H(Gn). Furthermore, if the groups G1, G2, . . . , Gnare cyclic groups, then H(G) is a trivial group. In this paper we show that for every group G there exists a group denoted H(G) and is called the associated group of G satisfying some important properties that as application we show that if F is a quasi-free group and G is any group, then H(F )is trivial and H(F ∗ G) ≊ H(G), where a group is termed a quasi-free group if it is a free product of cyclic groups of any order. 2020 Mathematics Subject Classifications: 16U60, 20C05, 16S34, 20E06 Key Words and Phrases: Free groups, cyclic groups, quasi-free group, free product of groups and associated groups. 1. Introduction We introduce the following basic concepts needed for the definition of associated groups of given groups [1]. (1) Let G be a group. i. If A and B are two subsets of G, let [A,B] be the subgroup of G generated by the elements [a, b] = aba−1b−1 with a ∈ A and b ∈ B. Define G′ = [G,G] to be the derived subgroup of G generated by the elements [x, y] = xyx−1y−1 with x, y ∈ G. It is clear that G′ is a normal subgroup of G. For more details see [8]. ii. If R is a subset of G, let RG to be the intersection of all normal subgroups of G containing R. It is clear that R ⊆ RG and RG is a normal subgroup of G [7]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5258 Email addresses: am.alotaibi@psau.edu.sa (A. M. Alotaibi), khaled aljammal@yahoo.com (K. M. Aljamal) https://www.ejpam.com 2329 © 2024 EJPAM All rights reserved. A. M. Alotaibi, K. M. Aljamal / Eur. J. Pure Appl. Math, 17 (3) (2024), 2329-2335 2330 (2) Let X be a set and let FX be the free group of the reduced words of X generated by X. Then any element f ∈ FX , f ̸= 1 is uniquely written as f = x1 α1χ2 α2 . . . xn αn, xα+1 i+1 ̸= −αi, xi ∈ X,αi = ±1, i = 1, 2, . . . , n. A group H is called a free group of base S ⊆ H if H is isomorphic to FS that is, H ∼= FS . The universal property of the free group F of base S [4], states that given any function f : S → G from S to a group G, there exists a unique homomorphism φ : FS → G called the universal extension of f , that is, if a ∈ S, then φ(a) = f(a). Also, φ is an epimorphism if and only if f(S) generates G. (3) Let X be a set and let R ⊆ FX a subset of FX . Let ⟨x | R⟩ stand for the quotient group, such that ⟨x | R⟩ = FX/R̄, where R̄ = RFX is the normal closure of R in FX .⟨x | R⟩ is called a presentation. We say that the group G has the presentation ⟨x | R⟩ if G ∼= ⟨x | R⟩. From above we see that a group G has the presentation ⟨x | R⟩ if and only if there exists an onto function f : X → G, such that f(X) generates G and the normal closure RFX of R in FX satisfies the condition that RFX = ker(φ), where φ : FX → G is the universal extension of f . 2. The Associated Groups The concept of the associated group of a given group is introduced in [3] and [6] is defined as follows. Let G denote an arbitrary group. For x, y ∈ G, let ⟨x, y⟩ and let ⟨G,G⟩ = {⟨x, y⟩ : x, y ∈ G} and F⟨G,G⟩ be the free group freely generated by all pairs ⟨x, y⟩ with x, y ∈ G. Then any element α ∈ F⟨G,G⟩, α ̸= 1 is uniquely written as α = ⟨x1, y1⟩α1 ⟨x2, y2⟩α2 . . . ⟨xn, yn⟩αn ⟨xi+1, yi+1⟩αi+1 ̸= ⟨xi, yi⟩−αi, where xi, yi ∈ G,αI = ±1, i = 1, 2, . . . , n. Proposition 1. For any group G there is a unique epimorphism from F⟨G,G⟩ to [G,G] taking each element ⟨x, y⟩ ∈ F⟨G,G⟩, x, y ∈ G to the element [x, y] = xyx−1y−1 ∈ [G,G]. Proof. Let f : ⟨G,G⟩ → [G,G] be the function given by f(⟨x, y⟩) = [x, y] = xyx−1y−1 with x, y ∈ G. Since F ⟨G,G⟩ is a free group on ⟨G,G⟩, the universal property shows that there exists a unique homomorphism the function φG : F⟨G,G⟩ → [G,G] satisfying the condition that ⟨x1, y1⟩φG(⟨x, y⟩) = [x, y] = xyx−1y−1 with x, y ∈ G. This shows that f is the restriction of φG on f . That is, φG | ⟨G,G⟩ = f , or φG(⟨x, y⟩) = f(⟨x, y⟩) for all,y ∈ G. So for any element α ∈ F⟨G,G⟩, α ̸= 1, α can be written uniquely as α = ⟨x1, y1⟩α1 ⟨x2, y2⟩α2 . . . ⟨xn, yn⟩αn. and the value of α under φG is given by φG(α) = ⟨x1, y1⟩α1 ⟨x2, y2⟩α2 . . . ⟨xn, yn⟩αn. Now if Φ : F⟨G,G⟩ → [G,G] is a homomorphism, such that Φ(⟨x, y⟩) = [x, y] = xyx−1y−1 with x, y ∈ G, then Φ = φG. Consequently, φG is the unique required homomorphism. A. M. Alotaibi, K. M. Aljamal / Eur. J. Pure Appl. Math, 17 (3) (2024), 2329-2335 2331 Since f(⟨G,G⟩) = [G,G] generates [G,G], this implies that φG is an epimorphism. This complete the proof. Definition 1. For any group G, let φG:F⟨G,G⟩ → [G,G] be the unique epimorphism of Proposition 1 satisfying the condition that φG(⟨x, y⟩) = [x, y] = xyx−1y−1 with x, y ∈ G. We denote by C(G) = KerφG ) the kernel of φG. Then it is clear that C(G) is a normal subgroup of F⟨G,G⟩. Definition 2. For any group G, let R(G) ⊆ F⟨G,G⟩ be the set of the following elements of ⟨x, x⟩ ⟨x, y⟩⟨y, x⟩ ⟨y, z⟩x⟨x, z⟩⟨xy, z⟩−1 ⟨y, z⟩x⟨y, z⟩−1⟨x, [y, z]⟩−1  for x, y, z ∈ G, where ⟨y, z⟩x = ⟨yx, zx⟩ = 〈 xyx−1, xzx−1 〉 . Lemma 1. Let G be a group of presentation ⟨X | R⟩. Then H(G) ∼= R̄∩[FX , FX ] / [ FX , R̄ ] . Proof. See [3]. Theorem 1. For any group G, the normal closure [R(G)]F ⟨G,G 〉 of R(G) in F⟨G,G⟩ is contained in C(G). Proof. First we show that R(G) ⊆ C(G). This is equivalent of showing that the value of any element α ∈ R(G) under the epimorphism φG : F⟨G,G⟩ → [G,G] equals φG(α) = 1, the identity element of G. (1) Let x ∈ G. Then ⟨x, x⟩ ∈ F ⟨G,G⟩ and φG(⟨x, x⟩) = [x, x] = xx−1x−1 = 1. (2) Let x, y, z ∈ G. Then the elements ⟨y, z⟩x, ⟨x, z⟩, ⟨xy, z⟩−1 and ( ⟨y, z⟩x⟨x, z⟩⟨xy, z⟩−1 are in F⟨G,G⟩ and φG (〈 y, zx⟨x, z⟩⟨xy, z⟩−1 ) = φG ( ⟨y, z⟩X ) φG(⟨x, z⟩)φG ( ⟨xy, z⟩−1 )′′ (3) Let x, y, z ∈ G. Then the elements ⟨y, z⟩x, ⟨y, z⟩−1, ⟨x, [y, z]⟩−1 and ⟨y, z⟩x⟨y, z⟩−1⟨x, [y, z]⟩−1 are in F⟨G,G⟩ and φG ( ⟨y, z⟩x⟨y, z⟩−1⟨x, [y , z]⟩−1 ) = φG (⟨y, z⟩x)φG ( ⟨y, z⟩−1 ) φG ( ⟨x, [y, z]⟩−1 ) From above we have R(G) ⊆ C(G). Since C(G) is a normal subgroup of F⟨G,G⟩, this implies that the normal closure [R(G)] F ⟨G,G⟩ of R(G) in F⟨G,G⟩ is contained in C(G). This complete the proof. Definition 3. [3] For any group G, let B(G) = [R(G)]F ⟨G,G⟩ be the normal closure of R(G) in F⟨G,G⟩ and H(G) be the group H(G) = C(G)/B(G) = {αB(G) : α ∈ C(G)}, the quotient group of the set of left cosets of B(G) in C(G).H(G) is called the associated group of the group G. Proposition 2. The associated group of any infinite cyclic group is trivial. A. M. Alotaibi, K. M. Aljamal / Eur. J. Pure Appl. Math, 17 (3) (2024), 2329-2335 2332 Proof. If G is an infinite cyclic, then G is generated by a single element g and G has the presentation G = ⟨x | ∅⟩ = ⟨X | R⟩, where X = {x} and R = ∅, the empty set. Then the normal closure RFX of R in FX is {1}, the identity subgroup of G. By Lemma 1, H(G) ∼= R̄ ∩ [FX , FX ] / [ FX , R̄ ] = {1}/{1} = {1}. Consequently, H(G) ∼= {1}. This completes the proof. Theorem 2. The associated group of the free product of two groups is the direct product of associated groups of the two groups. That is, if K and L are two groups, then H(K ∗ L) ∼= H(K)×H(L). Proof. See [3]. Remark 1. We have the following notes regarding Theorem 2, let K = ⟨X | R⟩ and L = ⟨Y | S⟩ be presentations of the groups K and L, such that X ∩ Y = ∅. By [5], K ∗ L has the presentation K ∗ L = ⟨X ∪ Y | R ∪ S⟩. The definition of the presentation implies that K = ⟨X | R⟩ = FX/R̄, where R̄ is the normal closure of R in FX , L = ⟨Y | S⟩ = FY /S̄, where S̄ is the normal closure of S in FY , and K ∗ L = ⟨X ∪ Y | R ∪ S⟩ = FX∪Y/X∪Y , where X ∪ Y = [R∪S]FX∪Y , normal closure of R∪S in FX∪Y . Lemma 1, implies that H(K) ∼= R̄ ∩ [FX , FX ] / [ FX , R̄ ] , H(L) ∼= S̄∩[FY , FY ] / [ FY , S̄ ] andH(K∗L) ∼= (R ∪ S)∩[FX∪Y , FX∪Y ] / [ FX∪Y , R ∪ S ] . Corollary 1. Consider the groups K and L of presentations K = ⟨x0, x1, . . . , xn+1 | r1, . . . , rn, x0⟩, and L = ⟨x0, x1, . . . , xn+1 | r1, . . . , rn⟩. Then H(K) ∼= H(L). Proof. By [5], L = K ∗ P is the free product of K and P , where P is an infinite cyclic group. By Theorem 1, H(L) ∼= H(K)×H(P ) and by Proposition 2 H(P ) is trivial. That is, H(P ) ∼= {1}. This implies that H(L) ∼= H(K)×{1} ∼= H(K). This complete the proof. 3. The Associated Groups of Quasi-Free Groups Recall that a group is termed a quasi-free group if it is a free product of cyclic groups of any order. In this section we show that if F is a quasi-free group and G is any group then the associated group H(F ) of F is trivial and the associated group H(F ∗G) of the free product F ∗G of F and G satisfies the condition H(F ∗G) ∼= H(G). First we introduce the following concept. If G is a finite group, the Schur multiplier of G introduced in [8, p. 14] is denoted by M(G) and is defined to be the second cohomology group H2 (G,C∗) of G, where C∗ is the set of nonzero complex numbers. Proposition 3. Let G be a finite group. Then H(G) ∼= M(G). Furthermore, if G has the presentation ⟨X | R⟩, where X has cardinality m and R has cardinality n, then H(G) ∼= {1} if m = n and H(G) is cyclic if n = m+ 1. Proof. If G is finite, then by [2] we have H(G) = M(G). If m = n, then by [3], M(G) = 1. Consequently, H(H) = 1. If n = m+ 1, then M(G) is cyclic. This implies that H(G) is cyclic. This complete the proof. A. M. Alotaibi, K. M. Aljamal / Eur. J. Pure Appl. Math, 17 (3) (2024), 2329-2335 2333 Lemma 2. The associated group of any cyclic group is trivial. Proof. Let G be any cyclic group. We need to show that H(G) is trivial. If G is an infinite cyclic group, then by Proposition 2, H(G) ∼= {1}. If G is a finite cyclic group of order n then G has the presentation G = ⟨x | xn⟩ of one generator x and one relater xn = 1. So the number of the generators of G is the same number of relaters = 1. Then Proposition 3, shows that H(G) ∼= {1}. This complete the proof. Theorem 3. The associated group of a quasi-free group is trivial. Proof. Let F be a quasi-free group and G be any group. Then H(F ) ∼= {1} and H(F ∗G) ∼= H(G). Let F be a quasi-free group. Then F can be written as F = C∞ ∗ C∞ ∗ . . . ∗ C∞︸ ︷︷ ︸ p− factors ∗Cα1 ∗ Cα2 ∗ . . . ∗ Cαn︸ ︷︷ ︸ q− factors , where C∞ stands for an infinite cyclic group and Cα1 , Cα2 , . . . , Cαn stand for finite cyclic groups of orders α1, α2, . . . , αn respec- tively. From Proposition 1, we have Lemma 2, implies that H(F ) ∼= (1)× (1)× . . .× (1)︸ ︷︷ ︸ p− factors × (1)× (1)× . . .× (1)︸ ︷︷ ︸ q− factors ∼= 1. This completes the proof. Corollary 2. Let Z = {. . . ,−3,−2,−1, 0, 1, 2, 3, . . .} be the group of integers and Zn = {0, 1, . . . , n− 1} be the group of integers modulo n. Then H(Z) ∼= {1} and H (Zn) ∼= {1}. Corollary 3. If K is a free group and G is any group, then H(K) ∼= {1}, H(F ⟨G,G⟩) ∼= {1} and H ( F⟨G,G⟩ ∗G ) ∼= H(G). As an example of a quasi-free group we have the following. Example 1. Let Z be the group of integers and G = PSL(2, Z) be the projective special linear group of degree 2 over Z. It is well known that[1] , G = A ∗ B, the free product of the cyclic groups A of order 2 , and the cyclic group B of order 3 defined G is a quasi-free group, and Theorem 3, shows that H(F ∗G) ∼= {1}. 4. The Associated Groups of Dihedral Groups and Quaternion Groups For the structures of dihedral groups and quaternion groups we refer the readers to [8]. Proposition 4. Let G be a dihedral group. Then (i) If G = D∞ is infinite, then H (D∞) ∼= {1}. (ii) If G = Dn is finite, then H (Dn) is cyclic. A. M. Alotaibi, K. M. Aljamal / Eur. J. Pure Appl. Math, 17 (3) (2024), 2329-2335 2334 Proof. (1) G = D∞ is defined as the group of two-by-two matrices with entries from the group of integers Z of the form( ε k 0 1 ) where E is 1 or -1 , and k is any integer. D∞ = A ∗B, the free product of the groups A = {( 1 0 0 1 ) , ( 1 1 0 1 )} , B = {( 1 0 0 1 ) , ( −1 0 0 1 )} . Aand B are of order 2 which implies A and B are cyclic groups, then D∞ is a quasi-free group and by Theorem 3.1, H (D∞) ∼= {1}. (2) If G = Dn is a finite dihedral group, then G is of order 2n and is defined as the group of two-by-two matrices, with entries from the ring of integers Zn mod n of the form ( ε k 0 1 ) , where ε is 1 or -1 , and k is any integer mod n. Then by [8], Dn has the presentation Dn = 〈 x, y | xn, y2, (xy)2 〉 of 2 generators and 3 relations. Since Dn is finite and 3 = 2+1, Proposition 3 shows that H (Dn) is cyclic. This complete the proof. Example 2. Let G = PSL(2, F ) be the projective special linear group of degree 2 over the Galois field F consisting of 5 elements. Then H(G) ∼= C2 and for every free group K have H(K ∗G) ∼= C2, where C2 is a cyclic group of order 2, because it is well known that [3]. Now by [2], G has the presentation G = 〈 x, y | x5, y3, (xy)2 〉 of two generators and three relaters. By Proposition 3, H(G) is cyclic. Proposition 5. Let Qn be the quaternion group of order 4n. Then H (Qn) is cyclic. Proof. It is well known in [4] that Qn has the presentation Qn = 〈 a, b | a2n = 1, b2 = an, b−1ab = a−1 〉 . So the presentation of Qn is of 2 generators 3 relations so, by Proposition 3, H (Qn) is cyclic. This complete the proof. 5. Conclusion It is stated and proven associated group of any cyclic group is trivial. and associated group of a quasi-free group is trivial. In future work, we must reach facts related to the following (i) Let G = G1∗AG2 be the free product of the groups G1 and G2 with an amalgamation subgroup A introduced in [5]. Find H(G) in terms of H (G1) , H (G2) and H(A). (ii) Let G be the HNN-group G = ⟨H, ti | rel(H), tiAiti−1 = Bi, i ∈ I⟩ of base H and associated pairs (Ai, Bi), i ∈ I of subgroups of H introduced in [1]. Find H(G) in terms of H(H), H(Ai) and H(Bi), i ∈ I. REFERENCES 2335 Acknowledgements This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1446). References [1] Khaled Mustafa Aljamal, Ahmad Termimi Ab Ghani, and Rasheed Mahmood Saleh. On preimages of the quasi-treed hnn groups. In 2021 International Conference on Information Technology (ICIT), 2021. [2] Warren Dicks and Martin John Dunwoody. Groups acting on graphs. Cambridge University Press, 1989. [3] Gregory Karpilovsky. The schur multiplier. Oxford University Press, Inc., 1987. 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