EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 956-968 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the isomorphism problem for central extensions II Noureddine Snanou Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fez, Morocco Abstract. In this paper, we study the isomorphism problem for central extensions. More precisely, in some new situations, we provide necessary and sufficient conditions for two central extensions to be isomorphic. We investigate the case when the quotient group is simple or purely non- abelian. Furthermore, we characterize isomorphisms leaving the quotient group invariant. Finally, we deal with isomorphisms of central extensions where the kernel group and the quotient group are isomorphic. 2020 Mathematics Subject Classifications: 20J05, 20E22, 20J06 Key Words and Phrases: Central extension, Isomorphism problem, Lower isomorphic, Upper isomorphic, (G)-isomorphic 1. Introduction The classification of groups in a certain class is one of the most classical problems in group theory. For groups with composition series, the Jordan–Hölder Theorem states that if we can list all simple groups, and solve the extension problem then we can construct and classify all groups. The classification of simple groups has been achieved in the finite case, hence we need to solve the extension problem. The extension problem for two groups G1 and G2 is the problem of finding all groups G with G1 as a normal subgroup of G, and the quotient group G/G1 isomorphic to G2. Such a group G is called an extension of G1 by G2 [6]. The classification of extensions with non-abelian kernel group may be found in many texts, but the famous references for these extensions are Schreier’s paper [7] and Eilenberg-Mac Lane’s paper [4]. In this work, we will focus on extensions with abelian kernel group. In particular, if G1 is a central subgroup of G, then we say that G is a central extension of G1 by G2. For central extensions, an answer to the extension problem has been given by Hölder and Schreier by using the group cohomology [6, Theorem 7.59]. However, this answer will not enable us to construct all possible non-isomorphic central extensions of G1 by G2 (the isomorphism problem). In fact, it is very hard to solve the isomorphism problem, but it has been discussed for some special cases in [8–10]. In fact, DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5118 Email address: noureddine.snanou@usmba.ac.ma (N. Snanou) https://www.ejpam.com 956 © 2024 EJPAM All rights reserved. N. Snanou / Eur. J. Pure Appl. Math, 17 (2) (2024), 956-968 957 those results do not concern general isomorphisms, but only those of certain type, namely leaving the kernel group or both the two factors invariant, inducing the identity or a commuting automorphism on the quotient group. In this work, necessary and sufficient conditions for two central extensions of G1 by G2 to be isomorphic are given in some new situations. More precisely, we study the case when the quotient group is simple or purely non-abelian. Furthermore, we characterize isomorphisms leaving the quotient group invariant, and deal with isomorphisms of central extensions where the kernel group G1 and the quotient group G2 are isomorphic. Throughout this paper, we denote by Z(G), G′, Aut(G) and End(G), respectively, the center, the derived subgroup, the automorphism group, and the monoid of endomorphisms of G. For any two groups H and K, let Hom(H,K) denote the set of all homomorphisms from H to K. 2. Central extension In this paper, aspects of group cohomology will be used frequently. Therefore, we recall in this section some basic facts of this theory and fix additional notations and terminology. Let 1 → G1→G→G2 → 1 be a group extension, where for convenience we regard the kernel group G1 as a subgroup of G and G2 is identified with the quotient group G/G1. If G1 is a central subgroup of G, then we say that G is a central extension of G1 by G2. Two central extensions G and G′ of G1 by G2 are said to be equivalent if there exists a homomorphism φ : G→ G′ such that the diagram 1 → G1 → G → G2 → 1 ∥ ↓ φ ∥ 1 → G1 → G′ → G2 → 1 commutes. LetG2 be a group which acts trivially on a groupG1. A 2-cocycle ofG2 with coefficients in G1 is a map ε : G2 ×G2 → G1 satisfying the 2-cocycle condition, that is ε(h, g)ε(hg, k) = ε(g, k)ε(h, gk) for all g, h, k ∈ G2. We always assume that ε is normalized, i.e. ε(g, 1) = ε(1, g) = 1 for all g ∈ G2. Note that 2-cocycles are known by factor sets in many books (see for example [1–3, 5, 6, 12]). The set of normalized 2-cocycles of G2 with coefficients in G1 is denoted by Z2(G2, G1). The trivial 2-cocycle is the 2-cocycle c with c(g, h) = 1 for all g, h ∈ G2. Let ε1, ε2 ∈ Z2(G2, G1). We write ε1 ∼ ε2 and say that ε1 and ε2 are cohomologous, if there is a map t : G2 → G1 such that ε2(g, h) = t(g)t(h)ε1(g, h)t(gh) −1 for all g, h ∈ G2. Then (∼) is an equivalence relation on Z2(G2, G1). The cohomology class of ε ∈ Z2(G2, G1) is denoted by [ε]. The set of all cohomology classes of G2 with coefficients in G1 is denoted by H2(G2, G1) and called the second cohomology of G2 with coefficients in G1. From now, G1 will always considered an abelian group. Then Z2(G2, G1) is an abelian group and we have H2(G2, G1) = Z2(G2, G1)/B 2(G2, G1) where B2(G2, G1) is the sub- group of Z2(G2, G1) which consists of all functions ψ : G2×G2 → G1 satisfying that for all N. Snanou / Eur. J. Pure Appl. Math, 17 (2) (2024), 956-968 958 g, h ∈ G2: ψ(h, g) = δ(g)δ(hg)−1δ(h) for some δ : G2 → G1. The elements of B2(G2, G1) are called 2-coboundaries. The set of all normalized 2-cocycles which are symmetric forms a subgroup of Z2(G2, G1) and denoted by SZ2(G2, G1). The famous Schreier theorem says that the central extensions of G1 by G2 are classified by the non-trivial elements of the second cohomology group H2(G2, G1) with coefficients in G1. In particular, a cen- tral extension of G1 by G2 splits if and only if the corresponding 2-cocycle is trivial in H2(G2, G1). A 2-cocycle ε ∈ Z2(G2, G1) gives rise to a central extension G = G1 × ε G2 of G1 by G2 induced by ε, with group operation given by (x, y) • ε (x′, y′) = (xx′ε(y, y′), yy′) for all x, x′ ∈ G1 and y, y′ ∈ G2. Conversely, given a central extension 1 → G1→G j→ G2 → 1 and choose a based section λ : G2 → G, i.e. a set map with λ(1) is the identity element of G and j◦λ = idG2 . The based section λ induces a 2-cocycle ελ : G2×G2 −→ G1 given by ελ(h, g) = λ(h)λ(g)λ(hg)−1 and therefore, the group G is isomorphic to the group G1 × ελ G2 [10, Proposition 2.3]. We can easily see that the group G1 × ε G2 is abelian if and only if G2 is abelian and ε ∈ SZ2(G2, G1). We know that G1 × ε G2 = G1 ×G2 if and only if ε = 1. But, it is possible for a central extension of G1 by G2 induced by a non-trivial 2-cocycle to be isomorphic to the direct product G1 ×G2 (see Proposition 3.2). 3. Preliminary results Let pri : G1 × ε G2 → Gi be the ith canonical projection and ti : Gi → G1 × ε G2 be the ith canonical injection. Let φ be a group homomorphism from G1 × ε1 G2 to G1 × ε2 G2 and set φij = pri ◦ φ ◦ tj , where 1 ≤ i, j ≤ 2. So we can write φ in the matrix form: φ = ( φ11 φ12 φ21 φ22 ) . Obviously, we see that pr2 and t1 are group homomorphisms, then φ21 is a group homomorphism. Furthermore, we have the following lemmas which we need in the sequel. Lemma 3.1. [10, Lemma 3.1] Let φ = ( φ11 φ12 φ21 φ22 ) be a group homomorphism from G1 × ε1 G2 to G1 × ε2 G2. Then φ(x, y) = (φ11(x)φ12(y)ε2(φ21(x), φ22(y)), φ21(x)φ22(y)) (1) for all x ∈ G1, and y ∈ G2. Lemma 3.2. [10, Lemma 3.2] Let φ be a set map from G1 × ε1 G2 to G1 × ε2 G2. Then φ is a group homomorphism if and only if N. Snanou / Eur. J. Pure Appl. Math, 17 (2) (2024), 956-968 959 φ(x, y) • ε2 φ(x′, 1) = φ(xx′, y), (2) and φ(x, y) • ε2 φ(1, y′) = φ(xε1(y, y ′), yy′) (3) for all x, x′ ∈ G1, and y, y ′ ∈ G2. Definition 3.1. Let G2 be a group which acts trivially on an abelian group G1. Let ε ∈ Z2(G2, G1). A map χ : G1 → G1 is called an ε-endomorphism of G1, if χ(xε(y, y′)) = χ(x)χ(ε(y, y′)) for all x ∈ G1, and y, y′ ∈ G2. If in addition χ is a bijection, then it is said to be ε-automorphism. The following lemma follows directly by using the 2-cocycle condition. Lemma 3.3. Let G2 be a group which acts trivially on an abelian group G1. Let ε ∈ Z2(G2, G1), δ ∈ Hom(G1, G2) and σ an ε-endomorphism of G1. Then σ ◦ε ∈ Z2(G2, G1), δ ◦ ε ∈ Z2(G2, G2) and ε ◦ (δ × δ) ∈ Z2(G1, G1). From now, if φ = ( φ11 φ12 φ21 φ22 ) is a map from G1 × ε1 G2 to G1 × ε2 G2, then φ is defined by the formula (1). From the previous Lemmas, we get the following interesting result which will be frequently used in the sequel. Proposition 3.1. Let G2 be a group such that the equivalence relation (∼) is trivial on Z2(G2, G2). Let φ = ( φ11 φ12 φ21 φ22 ) be a map from G1 × ε1 G2 to G1 × ε2 G2. Then, φ is a group homomorphism if and only if (i) φ21 ∈ Hom(G1, CG2(φ22(G2))), φ22 ∈ End(G2) and φ11 is an ε1-endomorphism, (ii) [{1} × φ22(G2), {1} × φ21(G1)] = 1, (iii) Im(ε1) ≤ Ker(φ21) and ε −1 2 ◦ (φ21 × φ21) = ψφ11 ∈ B2(G1, G1), (iv) (φ11 ◦ ε1)(ε−1 2 ◦ (φ22 × φ22)) = ψφ12 ∈ B2(G2, G1), where ψφij (y, y ′) = φij(y)φij(y ′)φij(yy ′)−1 for all 1 ≤ i, j ≤ 2 and y, y′ ∈ Gj. Proof. Indeed, evaluate the left hand side and right hand side of the formulas (2) and (3), we obtain φ(x, y) · ε2 φ(x′, 1) = φ(xx′, y) N. Snanou / Eur. J. Pure Appl. Math, 17 (2) (2024), 956-968 960 ⇔ (φ11(x)φ12(y)ε2(φ21(x), φ22(y)), φ21(x)φ22(y)) · ε2 (φ11(x ′), φ21(x ′)) = (φ11(xx ′)φ12(y)ε2(φ21(xx ′), φ22(y)), φ21(xx ′)φ22(y)) ⇔ φ21(x)φ22(y)φ21(x ′) = φ21(xx ′)φ22(y) and (4) φ11(x)φ12(y)ε2(φ21(x), φ22(y))φ11(x ′)ε2(φ21(x)φ22(y), φ21(x ′)) (5) = φ11(xx ′)φ12(y)ε2(φ21(xx ′), φ22(y)). Setting x = 1 in the equations (4) and (5), we obtain φ22(y)φ21(x ′) = φ21(x ′)φ22(y) (6) and ε2(φ22(y), φ21(x ′)) = ε2(φ21(x ′), φ22(y)). (7) That is, [{1} × φ22(G2), {1} × φ21(G1)] = 1. Now, combining the equations (4) and (6), we get that φ21 ∈ Hom(G1, CG2(φ22(G2))). Thus, using the 2-cocycle condition together with the equations (6) and (7), the equation (5) yields φ11(xx ′) = φ11(x)φ11(x ′)ε2(φ21(x), φ21(x ′)) (8) which implies that ε−1 2 ◦ (φ21 × φ21) ∈ B2(G1, G1). On the other hand, we have that φ(x, y) · ε2 φ(1, y′) = φ(xε1(y, y ′), yy′) ⇔ (φ11(x)φ12(y)ε2(φ21(x), φ22(y)), φ21(x)φ22(y)) · ε2 (φ12(y ′), φ22(y ′)) = (φ11(xε1(y, y ′))φ12(yy ′)ε2(φ21(xε1(y, y ′)), φ22(yy ′)), φ21(xε1(y, y ′))φ22(yy ′)) ⇔ φ21(x)φ22(y)φ22(y ′) = φ21(xε1(y, y ′))φ22(yy ′) and (9) φ11(x)φ12(y)ε2(φ21(x), φ22(y))φ12(y ′)ε2(φ21(x)φ22(y), φ22(y ′)) (10) = φ11(xε1(y, y ′))φ12(yy ′)ε2(φ21(xε1(y, y ′)), φ22(yy ′)). Since φ21 is a group homomorphism, the equation (9) implies that φ21(ε1(y, y ′)) = φ22(y)φ22(y ′)(φ22(yy ′))−1 for all y, y′ ∈ G2. So, φ21◦ε1 ∼ 1. But, by the assumption, there is no nontrivial 2-cocycle in Z2(G2, G2) that is cohomologous to the trivial 2-cocycle. So, we have φ21 ◦ ε1 = 1 and then φ22 ∈ End(G2). Furthermore, by using the 2-cocycle condition, the equation (10) gives us φ11(xε1(y, y ′))φ12(yy ′) = φ11(x)φ12(y)φ12(y ′)ε2(φ22(y), φ22(y ′)). (11) But, the equation (8) yields φ11(xε1(y, y ′)) = φ11(x)φ11(ε1(y, y ′)). Thus, the equation (11) is equivalent to φ11(ε1(y, y ′))φ12(yy ′) = φ12(y)φ12(y ′)ε2(φ22(y), φ22(y ′)), N. Snanou / Eur. J. Pure Appl. Math, 17 (2) (2024), 956-968 961 which implies that φ11(ε1(y, y ′))(ε2(φ22(y), φ22(y ′)))−1 = φ12(y)φ12(y ′)(φ12(yy ′))−1. Thus, the proof is completed. Remark 3.1. (i) Note that if G2 is abelian then the assumption of the previous propo- sition and the condition B2(G2, G2) = 1 are equivalent. (ii) If ε2 ∈ SZ2(G2, G1) then the second condition of the previous proposition is not required. Definition 3.2. The groups G1 × ε1 G2 and G1 × ε2 G2 are called upper isomorphic, if there exists an isomorphism φ : G1 × ε1 G2 −→ G1 × ε2 G2 leaving G1 invariant. Remark 3.2. It is possible for two central extensions to be isomorphic without being upper isomorphic. For example, consider the central extensions 1 → ZN p i1→ G→ZN p → 1 and 1 → ZN p i2→ G→ZN p → 1 such that G = Zp × (Zp2) N, i1(ZN p ) = {1} × (pZp2) N and i2(ZN p ) = Zp × (pZp2) N. We have i1(ZN p ) ∼= i2(ZN p ) ∼= ZN p and G/i1(ZN p ) ∼= G/i2(ZN p ) ∼= ZN p . Since i1(ZN p ) = pG is a characteristic subgroup of G and i2(ZN p ) ̸= pG, it follows that there is no automorphism of G sending i1(ZN p ) to i2(ZN p ). In particular, suppose that Z(G1× ε1 G2) = Z(G1× ε2 G2) = G1 or (G1× ε1 G2) ′ = (G1× ε2 G2) ′ = G1. So each isomorphism φ = ( φ11 φ12 φ21 φ22 ) between G1 × ε1 G2 and G1 × ε2 G2 leaves G1 invariant and then φ21 = 1. So, the equation (9) implies that φ22 ∈ End(G2) and then the assumption of the previous proposition is not required. This case is covered by the following result which can be viewed as a consequence of [8, Theorem 3.7]. Proposition 3.2. The groups G1 × ε1 G2 and G1 × ε2 G2 are upper isomorphic if and only if there exist σ ∈ Aut(G1) and ρ ∈ Aut(G2) such that (σ ◦ ε1)(ε−1 2 ◦ (ρ× ρ)) ∈ B2(G2, G1). Example 3.1. Define a function ε0 : Z2 p → Zp by ε0(i, j) = 0 if i+ j < p, and ε0(i, j) = 1 if i+ j ⩾ p for all 0 ⩽ i, j < p. This function is the 2-cocycle corresponding to the central extension 0 → Zp → Zp2 → Zp → 0 induced by the based section λ : Zp → Zp2 defined by λ(i mod p) = i mod p2 for all 0 ⩽ i < p. Now, let ε ∈ Z2(Zp,Zp) and mε = ∑p−1 k=0 ε(k, 1). By using the 2-cocycle condition, we can check that ε −mεε0 ∈ B2(Zp,Zp). So, putting G1 = G2 = Zp, σ = idG1 and ρ = idG2 in the preceding proposition, the groups Zp × ε Zp and Zp × mεε0 Zp are upper isomorphic. N. Snanou / Eur. J. Pure Appl. Math, 17 (2) (2024), 956-968 962 In view of the preceding proposition, it is possible for a central extension induced by a non-trivial 2-cocycle to be isomorphic to the direct product of the two factors group. More precisely, for a non-trivial 2-cocycle ε ∈ Z2(G2, G1), it is easy to deduce that the groups G1× ε G2 and G1×G2 are upper isomorphic if and only if there exists σ ∈ Aut(G1) such that σ ◦ ε ∈ B2(G2, G1). 4. Central extensions with simple or purely non-abelian quotient group Proposition 4.1. Let G2 be a simple non-abelian group which acts trivially on an abelian group G1. The groups G1 × ε1 G2 and G1 × ε2 G2 are isomorphic if and only if they are upper isomorphic. Proof. The if direction is clear. For the converse, assume that G1 × ε1 G2 and G1 × ε2 G2 are isomorphic by an isomorphism φ = ( φ11 φ12 φ21 φ22 ) . It follows from the equation φ(x, 1) • ε2 φ(1, y) = φ(1, y) • ε2 φ(x, 1) that [φ21(x), φ22(y)] = 1 for all x ∈ G1, and y ∈ G2. Now let g ∈ G2, there exists an element (x, y) ∈ G1 × ε1 G2 such that φ(x, y) = (1, g), that is g = φ21(x)φ22(y). So, for all h ∈ G1, we have gφ21(h)g −1 = φ21(x)φ22(y)φ21(h)φ22(y) −1φ21(x) −1 = φ21(x)φ21(h)φ21(x) −1 ∈ φ21(G1). Thus, φ21(G1) is a normal subgroup of G2. As G2 is simple non-abelian, then φ21(G1) is either trivial or G2. If φ21(G1) = G2, then φ21 is an epimorphism and therefore G2 is abelian, a contradiction. Hence, φ21 = 1 and then φ maps G1 to itself, as required. Recall that a non-abelian group which has no non-trivial abelian direct factor is said to be purely non-abelian. Theorem 4.1. Let G2 be a finite purely non-abelian group which acts trivially on a finite abelian group G1. Let φ ∈ {( σ η δ ρ )∣∣∣∣ σ ∈ Aut(G1), η ∈ Hom(G2, G1) δ ∈ Hom(G1, G2), ρ ∈ Aut(G2) } where σ, δ and ρ satisfy the conditions: (i) [{1} ×G2, {1} × δ(G1)] = 1, (ii) ε2 ◦ (δ × δ) = 1 and δ ◦ ε1 = 1, (iii) σ ◦ ε1 = ε2 ◦ (ρ× ρ). Then, φ is an isomorphism from G1 × ε1 G2 to G1 × ε2 G2. Proof. Indeed, the map φ is defined by the formula (1). By Proposition 3.1, the map φ is clearly a group homomorphism. Now, assume that φ(x, y) = 1. So δ(x)ρ(y) = 1 N. Snanou / Eur. J. Pure Appl. Math, 17 (2) (2024), 956-968 963 and then ρ(y) = δ(x−1), which implies that σ(x)η(y)ε2(δ(x), δ(x −1)) = 1. So, the first equation of the condition (ii) ensures that σ(x)η(y) = 1, and then x = σ−1(η(y−1)). Hence, ρ−1(δ(σ−1(η(y)))) = y. From the condition (i), we have [G2, δ(G1)] = 1, that is δ(G1) ≤ Z(G2). Hence, we have Ψ = ρ−1 ◦ δ ◦ σ−1 ◦ η ∈ Hom(G2, Z(G2)), and then ImΨ ⊴ G2. So, by Fitting’s Lemma, we have G2 ∼= KerΨ × ImΨ which contradicts to the fact that G2 is purely non-abelian. Thus y = 1 and then x = 1. Therefore, the map φ is injective, and then it is an isomorphism. Remark 4.1. The previous proposition will not be true if G2 is not purely non-abelian. Indeed, assume that G1 is a direct factor of G2 and let φ = ( idG1 φ12 φ21 idG2 ) be a map from G1 × ε1 G2 to G1 × ε2 G2 where φ12(x) = φ21(x) = x−1 for all x ∈ G1. So, using the formula (1), we obtain φ(x, x) = (ε2(φ21(x), φ21(x −1)), 1). Hence, by using the first equation of the condition (ii), we get φ(x, x) = (1, 1). Therefore, φ is not an isomorphism. 5. Lower isomorphism problem for central extensions Definition 5.1. Let G2 be a group which acts trivially on an abelian group G1. The groups G1 × ε1 G2 and G1 × ε2 G2 are called lower isomorphic if there exists an isomorphism φ : G1 × ε1 G2 −→ G1 × ε2 G2 leaving G2 invariant. Note that Remark 3.2 also shows that two central extensions can be isomorphic without being lower isomorphic. We now present the following main result of this section. Theorem 5.1. Let G2 be a group such that the equivalence relation (∼) is trivial on Z2(G2, G2). If the groups G1 × ε1 G2 and G1 × ε2 G2 are lower isomorphic then there exist ρ ∈ Aut(G2), δ ∈ Hom(G1, Z(G2)) and an ε1-automorphism σ such that (i) [{1} ×G2, {1} × δ(G1)] = 1, Im(ε1) ≤ Ker(δ), (ii) ε−1 2 ◦ (δ × δ) = ψσ ∈ B2(G1, G1) where ψσ(x, x ′) = σ(x)σ(x′)σ(xx′)−1 for all x, x′ ∈ G1, (iii) ε2 ◦ (ρ× ρ) = σ ◦ ε1. Proof. Suppose that G1 × ε1 G2 and G1 × ε2 G2 are isomorphic by an isomorphism φ =( φ11 1 φ21 φ22 ) . From Lemma 3.1, we have that φ(x, y) = (φ11(x)ε2(φ21(x), φ22(y)), φ21(x)φ22(y)) for all x ∈ G1, y ∈ G2. Since φ is bijective, so is φ22. By Proposition 3.1, the maps φ21 ∈ Hom(G1, Z(G2)), φ22 ∈ Aut(G2) and the ε1-endomorphism φ11 satisfy the con- ditions (i)-(iii). So, it remains to show that φ11 is bijective. Let g ∈ G1, since φ is N. Snanou / Eur. J. Pure Appl. Math, 17 (2) (2024), 956-968 964 surjective, there exists an element (x, y) ∈ G1 × ε1 G2 such that φ(x, y) = (g, 1), that is φ11(x)ε2(φ21(x), φ22(y)) = g and φ21(x)φ22(y) = 1. So, φ11(x)ε2(φ21(x), φ21(x −1)) = g and then, using the condition (ii), we have φ11(x) = g. Therefore, φ11 is surjective. On the other hand, the map ψ defined by ψ(x, y) = (x, yφ−1 22 (φ21(x) −1)) is a bijection and we have φ ◦ψ(x−1, 1) = (φ11(x −1)ε2(φ21(x −1), φ21(x)), 1) for all x ∈ G1. Thus, the condition (ii) ensures that φ ◦ψ(x−1, 1) = (φ11(x) −1, 1) for all x ∈ G1. Since φ ◦ψ is injective, so is φ11. Thus, the desired result follows directly by taking ρ = φ22, σ = φ11 and δ = φ21. Remark 5.1. The converse of the previous result holds if G1 and G2 are finite. Indeed, it suffices to show that φ is injective. Let (x, y) ∈ G1 ×ε G2 such that φ(x, y) = (1, 1). Then, the equality φ21(x)φ22(y) = 1 implies that φ22(y) = φ21(x −1). So φ11(x)ε2(φ21(x), φ21(x −1)) = 1. Using the condition (ii), we get φ11(x −1)−1 = 1 and then x = 1 since φ11 is injective. Since φ21(1) = 1, it follows that φ22(y) = 1 and then y = 1 since φ22 is injective. Therefore, φ is bijective and so it is a lower isomorphism by Proposition 3.1. As required. Let ε ∈ Z2(G2, G1) be a non-trivial 2-cocycle. In view of the preceding theorem, the group G1 × ε G2 cannot be lower isomorphic to the direct product G1 ×G2. Corollary 5.1. Further to the assumption of the previous theorem, suppose that B2(G1, G1) = 1. The groups G1 × ε1 G2 and G1 × ε2 G2 are lower isomorphic if and only if there exist ρ ∈ Aut(G2) and σ ∈ Aut(G1) such that ε2 ◦ (ρ× ρ) = σ ◦ ε1. Proof. Indeed, suppose that the groups G1× ε1 G2 and G1× ε2 G2 are lower isomorphic. By Theorem 5.1, there exist ρ ∈ Aut(G2), δ ∈ Hom(G1, Z(G2)) and an ε1-automorphism σ satisfying the conditions (ii) and (iii). Since B2(G1, G1) = 1, the condition (ii) implies that σ ∈ Aut(G1). Therefore, we conclude by using the condition (iii). For the converse, we can easily prove that the bijection φ defined by φ(x, y) = (σ(x), ρ(y)) is an isomorphism. Hence the corollary follows. Example 5.1. Define functions ε1, ε2 : (Z2 × Z2) 2 → Z2 by ε1((h1, h2), (g1, g2)) = h1g1 and ε2((h1, h2), (g1, g2)) = h2g2. These functions are 2-cocycles corresponding to two inequivalent central extensions of ⟨(2, 0)⟩ ∼= Z2 by Z2×Z2 whose the middle group is Z4×Z2. Note that ε1 and ε2 are the 2-cocycles induced by the based sections λ1, λ2 : Z2×Z2 → Z4× Z2 defined by λ1(h mod 2, g mod 2) = (h mod 4, g mod 2) and λ2(h mod 2, g mod 2) = (g mod 4, h mod 2). Let ρ ∈ Aut(Z2×Z2) defined by ρ(g1, g2) = (g2, g1) and take σ = idZ2, we can check easily that ε2 ◦ (ρ × ρ) = σ ◦ ε1. Therefore, the groups Z2 × ε1 (Z2 × Z2) and Z2 × ε2 (Z2 × Z2) are lower isomorphic. N. Snanou / Eur. J. Pure Appl. Math, 17 (2) (2024), 956-968 965 6. Isomorphisms of central extensions with isomorphic factors group Definition 6.1. Let G2 be a group which acts trivially on an abelian group G1. Let 1 ≤ i ≤ 2, the groups G1 × ε1 G2 and G1 × ε2 G2 are called (Gi)-isomorphic if there exists an isomorphism φ = ( φ11 φ12 φ21 φ22 ) between them such that φii = 1. Proposition 6.1. Let G2 be an abelian group and suppose that the groups G1 × ε1 G2 and G1× ε2 G2 are (G2)-isomorphic. Then, there exist an ε1-endomorphism σ of G1, an injective map η : G2 → G1 and an epimorphism δ : G1 → G2 such that: (i) ε−1 2 ◦ (δ × δ) = ψσ ∈ B2(G1, G1), Im(ε1) ≤ Ker(δ), (ii) σ ◦ ε1 = ψη ∈ B2(G2, G1). Proof. Let φ = ( φ11 φ12 φ21 φ22 ) be a (G2)-isomorphism from G1 × ε1 G2 to G1 × ε2 G2. So φ22 = 1, and therefore we can show easily that φ21 is surjective and φ12 is injective. Hence, by Proposition 3.1, the desired conditions follows directly by taking σ = φ11, δ = φ21 and η = φ12. Note that if G2 is non-abelian, then the condition δ ∈ Epi(G1, G2) implies that δ = 1. Therefore, the previous result becomes a direct consequence of Proposition 3.2. Corollary 6.1. Suppose that G1 and G2 are two finite abelian groups with the same order. The groups G1 × ε1 G2 and G1 × ε2 G2 are (G2)-isomorphic if and only if ε1 = 1 and there exists an isomorphism δ : G1 → G2 such that ε−1 2 ◦ (δ × δ) ∈ B2(G1, G1). Proof. Indeed, suppose that the groups G1× ε1 G2 and G1× ε2 G2 are (G2)-isomorphic. By the previous proposition, there exists an epimorphism δ : G1 → G2 such that Im(ε1) ≤ Ker(δ) and ε−1 2 ◦(δ×δ) ∈ B2(G1, G1). But δ is in fact an isomorphism by the assumption, so we must have ε1 = 1. Conversely, since ε−1 2 ◦ (δ× δ) ∈ B2(G1, G1), it follows that there exists a map σ : G1 → G1 such that ε−1 2 (δ(x), δ(x′)) = σ(x)σ(x′)σ(xx′)−1 for all x, x′ ∈ G1. Let δ′ be the inverse of δ. By the normalization condition, we have σ(1) = 1. So, the bijection φ defined by φ(x, y) = (σ(x)δ′(y), δ(x)) is clearly an isomorphism. As required. Proposition 6.2. Let G2 be a group such that the equivalence relation (∼) is trivial on Z2(G2, G2). Suppose that the groups G1 × ε1 G2 and G1 × ε2 G2 are (G1)-isomorphic. Then, there exist ρ ∈ End(G2), a surjective map η : G2 → G1 and a monomorphism δ : G1 → G2 such that (i) ε1 = 1, (ii) ε2 ◦ (δ × δ) = 1 and N. Snanou / Eur. J. Pure Appl. Math, 17 (2) (2024), 956-968 966 (iii) ε−1 2 ◦ (ρ× ρ) = ψη ∈ B2(G2, G1). Proof. Let φ = ( 1 φ12 φ21 φ22 ) be an isomorphism from G1 × ε1 G2 to G1 × ε2 G2. Hence, by taking ρ = φ22, η = φ12 and δ = φ21, the conditions (ii) and (iii) follow directly from Proposition 3.1. Notice that φ(x, y) = (φ12(y)ε2(φ21(x), φ22(y)), φ21(x)φ22(y)) for all x ∈ G1, y ∈ G2. So, φ(x, 1) = (1, φ21(x)) and then φ21 is injective. But, by the condition (iii) of Proposition 3.1, we have Im(ε1) ≤ Ker(φ21), which implies that ε1 = 1. On the other hand, let g ∈ G1, then there exists (x, y) ∈ G1 × ε1 G2 such that φ(x, y) = (g, 1). This gives us φ12(y)ε2(φ21(x), φ21(x −1)) = g. So the condition (ii) ensures that φ12(y) = g and then φ12 is surjective. As required. Now, we derive the following consequence. Corollary 6.2. Further to the assumption of the previous proposition, suppose that G1 and G2 are two finite abelian groups with the same order. The groups G1× ε1 G2 and G1× ε2 G2 are (G1)-isomorphic if and only if ε1 = 1 and there exists an isomorphism δ : G1 → G2 such that ε2 ◦ (δ × δ) = 1. Proof. Indeed, using the assumptions, the only if direction comes immediately from the previous result. Conversely, let δ′ be the inverse of δ. Define a bijective map φ between G1 × G2 and G1 × ε2 G2 given by φ(x, y) = (δ′(y), δ(x)), for all x ∈ G1, y ∈ G2. Since ε2 ◦ (δ × δ) = 1, it is easy to check that φ is a group homomorphism, and therefore it is a group isomorphism. 7. Conclusions and future problems As mentioned in the introduction, our choice to focus on the isomorphism problem for extensions with abelian kernel group is partially motivated by the Jordan–Hölder Theo- rem. Our other works of particular relevance are [8–11]. We study in [11] the isomorphism problem for split extensions, and as an application, we determine how isomorphism of split extensions and conjugacy of the images of the corresponding actions are related. As is well known, the identity element of the second cohomology group corresponds to the split extensions. This motivates us to consider also the non-identity different elements, which give rise to the construction of non-split extensions. So, in [8–10], we study the isomor- phism problem for non-split abelian extensions in some special cases. We mainly deal with isomorphisms leaving one of the two factors or even both invariant. We characterize such isomorphisms in various situations with some assumptions on the quotient and the kernel group. In this paper, by using a similar approach, we give a further contribution to this topic. More precisely, we complete the work with the isomorphism problem for central extensions in different special cases. So most of the results obtained here considered as a generalization of those in [9, 10]. REFERENCES 967 After the outcomes obtained in this work, one may naturally try to obtain similar results with suitable properties for G1 and G2 other than previously studied. One may also would like to study the isomorphism problem for non-split extensions with non-abelian kernel. This is of course the most important future problem on this topic. In our future works, we will continue with the isomorphism problem for extensions with abelian kernel group and try to generalize the results into any groups G1 and G2. We also would like to choose another groups extension such as Zappa-Szep product to apply similar results. Acknowledgements The author thanks the reviewers and the editor for their helpful comments and cor- rections, which have greatly improved the presentation of the paper. References [1] K. S. Brown. Cohomology of Groups. In Graduate Texts in Mathematics., volume 87. Springer-Verlag, New York, Heidelberg, Berlin, 1982. [2] D. S. Dummit and R. M. Foote. Abstract Algebra. John Wiley and Sons, Inc, New York, 3rd edition, 2004. [3] S. Eilenberg and S. MacLane. Group extensions and homology. Ann. of Math., Second Series, 43(4):757–831, 1942. [4] S. Eilenberg and S. MacLane. Cohomology Theory in Abstract Groups II. Ann. of Math., Second Series, 48(1):326–341, 1947. [5] S. MacLane. Homology. 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