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The identified subgroup used to form a central product is central, hence normal

Statement

Let G and H be groups with central subgroups Z1≤Z(G) and Z2≤Z(H) and an isomorphism α:Z1→Z2. The subgroup N={(z,α(z)−1):z∈Z1} of G×H is central, hence normal, so the quotient (G×H)/N of The central product G∘αH of two groups along an isomorphism of central subgroups is defined.

Facts & Assumptions

Given: Groups G,H, central subgroups Z1≤Z(G) and Z2≤Z(H), and an isomorphism α:Z1→Z2.

[F1]

For groups G,H with central subgroups Z1≤Z(G), Z2≤Z(H) and an isomorphism α:Z1→Z2, the central product G∘αH is the quotient of G×H by N={(z,α(z)−1):z∈Z1} (The central product G∘αH of two groups along an isomorphism of central subgroups).

[F2]

A subset H⊆G is a subgroup when e∈H, H is closed under the operation, and H is closed under inverses (Subgroup).

[F3]

Z(G):={z∈G:zg=gz for every g∈G} (The center Z(G) of a group).

[F4]

A subgroup N≤G is normal in G when gNg−1=N for every g∈G, where gNg−1:={gng−1:n∈N} (Normal subgroup: invariance under conjugation).

[L1]

A group homomorphism f satisfies f(e)=e′ and f(g−1)=f(g)−1 (A group homomorphism automatically satisfies f(e)=e′ and f(g−1)=f(g)−1, and f(gn)=f(g)n for every n∈Z; for monoid homomorphisms preservation of the identity must be assumed).

[L2]

The external direct product G×H:={(g,h):g∈G, h∈H} carries the componentwise operation (g,h)(g′,h′)=(gg′,hh′) (The external direct product G×H with componentwise multiplication).

[L3]

The componentwise operation makes G×H a group with identity (eG,eH) and (g,h)−1=(g−1,h−1) (G×H is a group with identity (eG,eH), coordinatewise inverses, and homomorphic coordinate projections).

Proof

technique · direct
1.1F3L1algebra

The isomorphism α is in particular a homomorphism, so α(e)=e and α(z−1)=α(z)−1; moreover Z2≤Z(H), so any two elements of Z2 commute and α(z1z2)−1=α(z1)−1α(z2)−1.

2.1F1F2L2L3step 1.1

Hence (e,e)=(e,α(e)−1) lies in N; the product (z1,α(z1)−1)(z2,α(z2)−1)=(z1z2,α(z1)−1α(z2)−1)=(z1z2,α(z1z2)−1) lies in N; and (z,α(z)−1)−1=(z−1,α(z))=(z−1,α(z−1)−1) lies in N. So N is a subgroup of G×H.

3.1F1F3L2step 2.1

Every element of N has first coordinate in Z1≤Z(G) and second coordinate in Z2≤Z(H), and the operation is componentwise, so (z,α(z)−1)(g,h)=(zg,α(z)−1h)=(gz,hα(z)−1)=(g,h)(z,α(z)−1) for every (g,h); thus N≤Z(G×H).

4.1F4step 3.1∎

For n∈N and x∈G×H centrality gives xnx−1=nxx−1=n, so xNx−1=N and N is normal; the quotient (G×H)/N is therefore defined.

Remarks

Centrality is used twice. It makes the inverse-coordinate rule z↦(z,α(z)−1) multiplicative, so that N is a subgroup, and it then makes that subgroup central and hence normal. For merely isomorphic subgroups the displayed antidiagonal need be neither a subgroup nor a normal subset, so the quotient construction does not apply.

Depends on

Used by

Cited to discharge well-definedness by The central product G∘_α H of two groups along an isomorphism of central subgroups.

Dependency tree · two levels

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Sources