Schur Multiplier of Central Product of Groups

Home , Central product

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 = {(a, φ(a)−1) | a ∈ A}. 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 = {(a, φ(a)−1) | a ∈ A}. 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 (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, ψ), deﬁned by Iwahori, Matsumoto, where ψ(ξ)(¯x, n) = f(x, n) − f(n, x) forx ¯ = xX0 ∈ X/X0 and n ∈ N, where ξ ∈ 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 ∈ H2(G, D) is represented by a 2-cocycle f, then ν(ξ) is the homomorphism f¯ ∈ 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 ∈ H2(G, D) is represented by a 2-cocycle f, then ν(ξ) is the homomorphism f¯ ∈ 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 ∈ H2(G, D) is represented by a 2-cocycle f, then ν(ξ) is the homomorphism f¯ ∈ 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,ψ) (res,ψ),(res,ψ),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) = {inf(η) | η ∈ θ−1 Im(tra, tra, 0)}.

1 Set Z = H0 ∩ K0, where X0 denotes the commutator subgroup of a group X.

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 14 / 28 Results Results

Lemma

Ker(θ0) = {inf(η) | η ∈ θ−1 Im(tra, tra, 0)}.

1 Set Z = H0 ∩ K0, where X0 denotes the commutator subgroup of a group X.

2 We have an exact sequence

0 → H0 ∩ K0 −→α1 (A ∩ H0) ⊕ (A ∩ K0) −→α2 A ∩ G0 → 0,

which induces an exact sequence

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 14 / 28 −1 ∗ 2 Deﬁne χ(f) = inf ◦θ ◦ (tra, tra, 0)(g) such that f = α1(g).

Theorem

The following sequence is exact:

χ 0 0 → Hom(Z,D) −→ H2(G, D) −→θ H2(H,D)⊕H2(K,D)⊕Hom(H⊗K,D).

Results

α∗ α∗ 0→Hom(A∩G0,D)−→2 Hom(A∩H0,D)⊕Hom(A∩K0,D)−→1 Hom(Z,D) → 0,

∗ α2 is the homomorphism (res, res).

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 15 / 28 Results

α∗ α∗ 0→Hom(A∩G0,D)−→2 Hom(A∩H0,D)⊕Hom(A∩K0,D)−→1 Hom(Z,D) → 0,

∗ α2 is the homomorphism (res, res). −1 ∗ 2 Deﬁne χ(f) = inf ◦θ ◦ (tra, tra, 0)(g) such that f = α1(g).

Theorem

The following sequence is exact:

χ 0 0 → Hom(Z,D) −→ H2(G, D) −→θ H2(H,D)⊕H2(K,D)⊕Hom(H⊗K,D).

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 15 / 28 Results

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,ψ) (res,ψ),(res,ψ),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 16 / 28

1 Corollary (Blackburn, Evens, 1979) Let G be an extra-special p-group of order p2n+1, n ≥ 2. Then M(G) is an elementary abelian p-group of order p2n2−n−1.

Results

Theorem A

Let B be a subgroup of G such that B ≤ Z. Then

H2(G, D) =∼ H2(G/B, D)/N, where N =∼ Hom(B,D).

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 17 / 28 Results

Theorem A

Let B be a subgroup of G such that B ≤ Z. Then

H2(G, D) =∼ H2(G/B, D)/N, where N =∼ Hom(B,D).

Corollary (Blackburn, Evens, 1979) Let G be an extra-special p-group of order p2n+1, n ≥ 2. Then M(G) is an elementary abelian p-group of order p2n2−n−1.

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 17 / 28 Results

0 1 L =∼ Hom (A ∩ H )/Z, D

0 2 M =∼ Hom (A ∩ K )/Z, D

2 2 3 Consider inf : H (G/A, D) → H (G/Z, D). Then

Im(inf) =∼ H2(H/A, D)/L ⊕ H2(K/A, D)/M ⊕ Hom(H/A ⊗ K/A, D)

embeds in H2(G/Z, D).

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 18 / 28 Theorem

Hom(Z,D) embeds in H2(H/A, D)/L ⊕ H2(K/A, D)/M.

Corollary Hom(Z,D) embeds in H2(H/Z, D) ⊕ H2(K/Z, D).

Results

• H2(G, D) ∼= H2(G/Z, D)/N, where N ∼= Hom(Z,D).

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 19 / 28 Corollary Hom(Z,D) embeds in H2(H/Z, D) ⊕ H2(K/Z, D).

Results

• H2(G, D) ∼= H2(G/Z, D)/N, where N ∼= Hom(Z,D). Theorem

Hom(Z,D) embeds in H2(H/A, D)/L ⊕ H2(K/A, D)/M.

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 19 / 28 Results

• H2(G, D) ∼= H2(G/Z, D)/N, where N ∼= Hom(Z,D). Theorem

Hom(Z,D) embeds in H2(H/A, D)/L ⊕ H2(K/A, D)/M.

Corollary Hom(Z,D) embeds in H2(H/Z, D) ⊕ H2(K/Z, D).

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 19 / 28 Results

Theorem B

Let L, M be deﬁned as above and N =∼ Hom(Z,D). Then the following statements hold true: (i) H2(H/A, D)/L ⊕ H2(K/A, D)/M/N ⊕ Hom(H/A ⊗ K/A, D) embeds in H2(G, D). (ii) H2(G, D) embeds in H2(H/Z, D) ⊕ H2(K/Z, D)/N ⊕ Hom(H ⊗ K,D).

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 20 / 28 2 2 2 Im(inf) =∼ H (H/A, D)/L ⊕ H (K/A, D)/M ⊕ Hom(H/A ⊗ K/A, D) embeds in H2(G/Z, D).

2 3 H (G/Z, D) embeds in H2(H/Z, D)/L ⊕ H2(K/Z, D)/M ⊕ Hom(H ⊗ K,D).

Results

2 2 1 H (G, D) =∼ H (G/Z, D)/N, where N =∼ Hom(Z,D).

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 21 / 28 2 3 H (G/Z, D) embeds in H2(H/Z, D)/L ⊕ H2(K/Z, D)/M ⊕ Hom(H ⊗ K,D).

Results

2 2 1 H (G, D) =∼ H (G/Z, D)/N, where N =∼ Hom(Z,D).

2 2 2 Im(inf) =∼ H (H/A, D)/L ⊕ H (K/A, D)/M ⊕ Hom(H/A ⊗ K/A, D) embeds in H2(G/Z, D).

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 21 / 28 Results

2 2 1 H (G, D) =∼ H (G/Z, D)/N, where N =∼ Hom(Z,D).

2 2 2 Im(inf) =∼ H (H/A, D)/L ⊕ H (K/A, D)/M ⊕ Hom(H/A ⊗ K/A, D) embeds in H2(G/Z, D).

2 3 H (G/Z, D) embeds in H2(H/Z, D)/L ⊕ H2(K/Z, D)/M ⊕ Hom(H ⊗ K,D).

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 21 / 28 Examples Examples

• Neither of the two embeddings of Theorem B is an isomorphism.

3 1 Example1: Let H be the extraspecial p-groups of order p (n+1) and exponent p and K = Zp , where n ≥ 1. Let G be a ∼ 0 ∼ central product of H and K amalgamated at A = H = Zp. (n) Note that G = H × Zp . It is easy to see that 1 n(n+3)+2 ∼ 2 M(G) = Zp .

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 22 / 28 Examples

• First embedding in Theorem B can very well be an isomorphism, but the second one can still be strict (i.e., not an isomorphism).

1 Example 2. Consider the group G presented as

p2 p p p G = hα, α1, α2, γ | [α1, α] = γ = α2, α = α1 = α2 = 1i.

2 Example 3. Consider the group G presented as p p G = hα, α1, α2, α3, γ | [α1, α] = α2, [α2, α] = γ = α3, α = (p) αi = 1, i = 1, 2, 3i.

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 23 / 28 Examples

• Both the embeddings in Theorem B can be isomorphisms.

3 1 Example 4. Let H be the extraspecial p-groups of order p 2 ∼ and exponent p and K = Zpn+1 , the cyclic group of order pn+1, where n ≥ 1. Let G be a central product of H and K ∼ 0 ∼ amalgamated at A = H = Zp.

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 24 / 28 Examples

Theorem If the second embedding in Theorem B is an isomorphism, then so is the ﬁrst.

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 25 / 28 Examples

Corollary (i) If A = Z, then

H2(G, D)=∼ H2(H/Z, D)⊕H2(K/Z, D)/ Hom(Z,D)⊕Hom(H/Z⊗K/Z, D).

(ii) If inf : H2(H/A, D) → H2(H/Z, D) and inf : H2(K/A, D) → H2(K/Z, D) are epimorphisms, then

H2(G, D)=∼ H2(H/Z, D)⊕H2(K/Z, D)/ Hom(Z,D)⊕Hom(H/A⊗K/A, D).

More precisely, the ﬁrst embedding in Theorem B is an isomorphism.

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 26 / 28 References References

Eckmann, B., Hilton, P. J. and Stammbach, U. On the Schur multiplicator of a central quotient of a direct product of groups, J. Pure Appl. Algebra 3 (1973), 73-82. Wiegold, J., Some groups with non-trivial multiplicators, Math. Z. 120 (1971), 307-308. Iwahori, N. and Matsumoto, H., Several remarks on projective representations of ﬁnite groups, J. Fac. Sci. Univ. Tokyo, Sect. I, 10 (1964), 129-146. Vermani, L. R., An exact sequence, Bull. London Math Soc., 6 (1974), 349-353.

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 27 / 28 References

Thank you for your attention!

Sumana Hatui (HRI) Groups St Andrews 2017 August 5-13, 2017 28 / 28