Lagmental Vicfred

Commutators Build the Largest Abelian Quotient by Vicfred

Last updated: Tue 15 June 2021

tags: mathematics, group theory, homomorphisms, quotients

Quotienting by the commutator subgroup removes exactly the obstruction to commutativity. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Set-up

A homomorphism \(\varphi:G\to H\) packages a comparison of operations. Its kernel \(\ker\varphi\) measures collapse, while its image \(\operatorname{im}\varphi\) records the part of \(H\) actually reached.

$$ [G,G]=\langle xyx^{-1}y^{-1}:x,y\in G\rangle $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ G^{\mathrm{ab}}=G/[G,G] $$

The calculation

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ \boxed{\ \operatorname{Hom}(G,A)\cong\operatorname{Hom}(G^{\mathrm{ab}},A)\ }\qquad(A\ \text{abelian}) $$

What survives abstraction

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad [G,G]=\langle xyx^{-1}y^{-1}:x,y\in G\rangle,\\[5pt] \mathsf{C}\;&:\quad G^{\mathrm{ab}}=G/[G,G]. \end{aligned} $$

The boundary

The quotient notation \(G/N\) is legal only for \(N\trianglelefteq G\). A set of cosets may exist without inheriting a well-defined group multiplication.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] G^{\mathrm{ab}}=G/[G,G] \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Thu 03 December 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.