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LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-09
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Commutator product identities in the fixed convention

Statement

Use [x,y]=xyx−1y−1 and uv=uvu−1. Then [x,yz]=[x,y]y[x,z] and [xy,z]=x[y,z][x,z]. Whenever the relevant commutators are central, these pairings are multiplicative in both variables and [xa,yb]=[x,y]ab for all a,b∈Z. Define [x1,…,xk]=[ [x1,…,xk−1],xk], with [x1]=x1.

Facts & Assumptions

Given: x,y,z are elements of a group; power assertions assume the displayed commutators are central.

[F1]

The commutator convention is [x,y]=xyx−1y−1 (Subgroup commutators and the lower central series).

Proof

1.1

[x,y]y[x,z]y−1=xyx−1y−1yxzx−1z−1y−1=xyzx−1z−1y−1=[x,yz].

F1algebra
1.2

x[y,z]x−1[x,z]=xyzy−1z−1x−1xzx−1z−1=xyzy−1x−1z−1=[xy,z]. Also [x,y]−1=yxy−1x−1=[y,x].

F1algebra
2.1

If commutators are central, their conjugates in steps 1.1 and 1.2 are unchanged. Each variable then defines a homomorphism on any subgroup where that centrality hypothesis holds. The identity has commutator 1, and 1=[xx−1,y]=[x,y][x−1,y] gives the inverse rule. Repeated multiplication gives positive powers, the identity gives exponent zero, and the inverse rule gives negative powers. Applying this in both variables yields [xa,yb]=[x,y]ab.

step 1.1step 1.2algebra∎

Source notes

Druţu–Kapovich, Lectures on Geometric Group Theory (585-page draft), Lemma 10.25 and Notation 10.26, printed p.281. The source product identities are expanded here in the fixed convention, including zero and negative exponents.

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Sources