Schur Multiplier of Central Product of Groups Sumana Hatui (Joint work with Manoj K. Yadav and L.R.Vermani) Harish-Chandra Research Institute (HRI), India Groups St Andrews 2017 University of Birmingham Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 1 / 28 Overview 1 Introduction 2 Preliminaries 3 Results 4 Examples 5 References Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 2 / 28 Introduction 1 Introduction 2 Preliminaries 3 Results 4 Examples 5 References Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 3 / 28 2 2 H (G; D): the second cohomology group of G with coefficients in D. Here D is a divisible abelian group regarded as a trivial G-module. ∼ 2 × ∗ 3 H2(G; Z) = (H (G; C )) is known as Schur multiplier of G. Introduction 1 H2(G; D): the second homology group of a group G with coefficients in D. Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 4 / 28 ∼ 2 × ∗ 3 H2(G; Z) = (H (G; C )) is known as Schur multiplier of G. Introduction 1 H2(G; D): the second homology group of a group G with coefficients in D. 2 2 H (G; D): the second cohomology group of G with coefficients in D. Here D is a divisible abelian group regarded as a trivial G-module. Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 4 / 28 Introduction 1 H2(G; D): the second homology group of a group G with coefficients in D. 2 2 H (G; D): the second cohomology group of G with coefficients in D. Here D is a divisible abelian group regarded as a trivial G-module. ∼ 2 × ∗ 3 H2(G; Z) = (H (G; C )) is known as Schur multiplier of G. Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 4 / 28 Definition (External central product) Let H, K be two groups with isomorphic subgroups A ≤ Z(H), B ≤ Z(K) under an isomorphism φ : A ! B. Consider the normal subgroup U = f(a; φ(a)−1) j a 2 Ag. Then the group G := (H × K)=U is called the external central product of H and K amalgamating A and B via φ. Introduction Definition Definition (Internal central product) Let G be group. A group G is called internal central product of its two normal subgroups H and K amalgamating A if G = HK with A = H \ K and [H; K] = 1. Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 5 / 28 Introduction Definition Definition (Internal central product) Let G be group. A group G is called internal central product of its two normal subgroups H and K amalgamating A if G = HK with A = H \ K and [H; K] = 1. Definition (External central product) Let H, K be two groups with isomorphic subgroups A ≤ Z(H), B ≤ Z(K) under an isomorphism φ : A ! B. Consider the normal subgroup U = f(a; φ(a)−1) j a 2 Ag. Then the group G := (H × K)=U is called the external central product of H and K amalgamating A and B via φ. Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 5 / 28 2 (K.Tahara, 1972) If G is the semidirect product of a normal subgroup H and a subgroup K, then H2(G; D) =∼ H2(K; D) × Hc2(G; D), where Hc2(G; D) is the kernel of res: H2(G; D) ! H2(K; D). 3 Let G be a central product of two groups H and K. We study H2(G; D), in terms of the second cohomology groups of certain quotients of H and K. Introduction 1 (I. Schur, 1907) H2(H × K; D) =∼ H2(H; D) × H2(K; D) × Hom(H ⊗ K; D). Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 6 / 28 3 Let G be a central product of two groups H and K. We study H2(G; D), in terms of the second cohomology groups of certain quotients of H and K. Introduction 1 (I. Schur, 1907) H2(H × K; D) =∼ H2(H; D) × H2(K; D) × Hom(H ⊗ K; D). 2 (K.Tahara, 1972) If G is the semidirect product of a normal subgroup H and a subgroup K, then H2(G; D) =∼ H2(K; D) × Hc2(G; D), where Hc2(G; D) is the kernel of res: H2(G; D) ! H2(K; D). Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 6 / 28 Introduction 1 (I. Schur, 1907) H2(H × K; D) =∼ H2(H; D) × H2(K; D) × Hom(H ⊗ K; D). 2 (K.Tahara, 1972) If G is the semidirect product of a normal subgroup H and a subgroup K, then H2(G; D) =∼ H2(K; D) × Hc2(G; D), where Hc2(G; D) is the kernel of res: H2(G; D) ! H2(K; D). 3 Let G be a central product of two groups H and K. We study H2(G; D), in terms of the second cohomology groups of certain quotients of H and K. Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 6 / 28 Theorem (Eckmann, Hilton and Stammbach, 1973) Let W be central in A = H × K with quotient G. Let U and V be the images of W under the projection of A onto H and K respectively. Then H=U ⊗ K=V is a quotient of H2(G; Z). Introduction Theorem (Wiegold, 1971) Let H; K be finite groups, let U; V be isomorphic central subgroups of H; K respectively, and let φ be an isomorphism from U onto V . Then the multiplicator of the central product G of H and K amalgamating U with V according to φ contains a subgroup isomorphic with H=U ⊗ K=V . Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 7 / 28 Introduction Theorem (Wiegold, 1971) Let H; K be finite groups, let U; V be isomorphic central subgroups of H; K respectively, and let φ be an isomorphism from U onto V . Then the multiplicator of the central product G of H and K amalgamating U with V according to φ contains a subgroup isomorphic with H=U ⊗ K=V . Theorem (Eckmann, Hilton and Stammbach, 1973) Let W be central in A = H × K with quotient G. Let U and V be the images of W under the projection of A onto H and K respectively. Then H=U ⊗ K=V is a quotient of H2(G; Z). Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 7 / 28 2 Then we have the exact sequence, 0 ! Hom(N \ X0;D) !tra H2(X=N; D) !inf H2(X; D) χ ! H2(N; D) ⊕ Hom(X ⊗ N; D); Preliminaries Preliminaries 1 Consider the following central exact sequence for an arbitrary group X and its central subgroup N: 1 ! N ! X ! X=N ! 1: Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 8 / 28 Preliminaries Preliminaries 1 Consider the following central exact sequence for an arbitrary group X and its central subgroup N: 1 ! N ! X ! X=N ! 1: 2 Then we have the exact sequence, 0 ! Hom(N \ X0;D) !tra H2(X=N; D) !inf H2(X; D) χ ! H2(N; D) ⊕ Hom(X ⊗ N; D); Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 8 / 28 Preliminaries 1 tra : transgression homomorphism 2 inf : inflation homomorphism 3 res : restriction homomorphism 4 χ = (res; ), defined by Iwahori, Matsumoto, where (ξ)(¯x; n) = f(x; n) − f(n; x) forx ¯ = xX0 2 X=X0 and n 2 N, where ξ 2 H2(X; D) and f is a 2-cocycle representative of ξ. Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 9 / 28 2 2 ν :H (G; D) ! Hom(H ⊗ K; D) is a homomorphism defined as follows: If ξ 2 H2(G; D) is represented by a 2-cocycle f, then ν(ξ) is the homomorphism f¯ 2 Hom(H ⊗ K; D) defined by f¯(h¯ ⊗ k¯) = f(h; k) − f(k; h); where h¯ = hH0 and k¯ = kK0. 3 θ0 is indeed a homomorphism. Preliminaries 1 Define a map θ0 :H2(G; D) ! H2(H; D) ⊕ H2(K; D) ⊕ Hom(H ⊗ K; D) θ0 = (res; res; ν) Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 10 / 28 3 θ0 is indeed a homomorphism. Preliminaries 1 Define a map θ0 :H2(G; D) ! H2(H; D) ⊕ H2(K; D) ⊕ Hom(H ⊗ K; D) θ0 = (res; res; ν) 2 2 ν :H (G; D) ! Hom(H ⊗ K; D) is a homomorphism defined as follows: If ξ 2 H2(G; D) is represented by a 2-cocycle f, then ν(ξ) is the homomorphism f¯ 2 Hom(H ⊗ K; D) defined by f¯(h¯ ⊗ k¯) = f(h; k) − f(k; h); where h¯ = hH0 and k¯ = kK0. Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 10 / 28 Preliminaries 1 Define a map θ0 :H2(G; D) ! H2(H; D) ⊕ H2(K; D) ⊕ Hom(H ⊗ K; D) θ0 = (res; res; ν) 2 2 ν :H (G; D) ! Hom(H ⊗ K; D) is a homomorphism defined as follows: If ξ 2 H2(G; D) is represented by a 2-cocycle f, then ν(ξ) is the homomorphism f¯ 2 Hom(H ⊗ K; D) defined by f¯(h¯ ⊗ k¯) = f(h; k) − f(k; h); where h¯ = hH0 and k¯ = kK0. 3 θ0 is indeed a homomorphism. Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 10 / 28 ∗ 0 / Hom(H=A ⊗ K=A; D) λ / Hom(H ⊗ K; D) µ∗ Hom(H ⊗ A; D) ⊕ Hom(K ⊗ A; D) Preliminaries • We have the exact sequence µ (H ⊗ A) ⊕ (K ⊗ A) −! H ⊗ K −!λ H=A ⊗ K=A ! 0: which induces the exact sequence Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 11 / 28 Preliminaries • We have the exact sequence µ (H ⊗ A) ⊕ (K ⊗ A) −! H ⊗ K −!λ H=A ⊗ K=A ! 0: which induces the exact sequence ∗ 0 / Hom(H=A ⊗ K=A; D) λ / Hom(H ⊗ K; D) µ∗ Hom(H ⊗ A; D) ⊕ Hom(K ⊗ A; D) Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 11 / 28 Preliminaries 2 1 X1 = H (A; D) ⊕ Hom(H ⊗ A; D) 2 2 X2 = H (A; D) ⊕ Hom(K ⊗ A; D) 3 X3 = Hom(H ⊗ A; D) ⊕ Hom(K ⊗ A; D) 2 2 4 Y = H (A; D) ⊕ H (A; D) Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 12 / 28 Preliminaries 0 0 (res;res) Hom(A \ G0) −−−−−! Hom(A \ H0) ⊕ Hom(A \ K0) tra (tra;tra;0) H2(G=A) −−−−−!θ H2(H=A) ⊕ H2(K=A) ⊕ Hom(H=A ⊗ K=A) inf (inf;inf,λ∗) 0 H2(G) −−−−−!θ H2(H) ⊕ H2(K) ⊕ Hom(H ⊗ K) (res; ) (res; );(res; ),µ∗ 2 H (A) ⊕ Hom(G ⊗ A) X1 ⊕ X2 ⊕ X3 ∼ ∗ ∗ = (4,α ,α ) * Y ⊕ X3 ⊕ X3: Diagram 1 Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 13 / 28 1 2 We have an exact sequence 0 ! H0 \ K0 −!α1 (A \ H0) ⊕ (A \ K0) −!α2 A \ G0 ! 0; which induces an exact sequence Results Results Lemma Ker(θ0) = finf(η) j η 2 θ−1 Im(tra; tra; 0)g.