Abelianization

The abelianization $𝐺 a b$ of a group $𝐺$ is the largest abelian quotient of $𝐺$ . group For a group $𝐺$ is abelianized by taking the quotient with the Commutator subgroup $[𝐺, 𝐺]$ .

𝐺 a b = 𝐺 / [𝐺, 𝐺]

Main theorem

Let $𝑁 ⊴ 𝐺$ . The quotient group $𝐺 / 𝑁$ is abelian iff $[𝐺, 𝐺] ⊴ 𝑁$ . group

Proof

$𝐺 / 𝑁$ is abelian iff $[𝑎, 𝑏] 𝑁 = 𝑎 𝑏 𝑎 - 1 𝑏 - 1 𝑁 = 𝑒 𝑁 = 𝑁$ for all $𝑎, 𝑏 \in 𝐺$ , and the latter holds iff $[𝑎, 𝑏] \in 𝑁$ for all $𝑎, 𝑏 \in 𝐺$ .

Universal property

Abelianization has a unique extension to a functor $(-) a b : 𝖦 𝗋 𝗉 \to 𝖠 𝖻$ from Category of groups into Category of abelian groups so that the projection becomes a natural transformation $𝜋 : 1 \Rightarrow (-) a b : 𝖦 𝗋 𝗉 \to 𝖦 𝗋 𝗉$ . This is done by defining $(𝐺 a b, 𝜋 𝐺)$ with the help of the following universal property:

$𝐺 a b$ is abelian. If $𝐻$ is an abelian group and $𝜑 \in 𝖦 𝗋 𝗉 (𝐺, 𝐻)$ is a homomorphism, then there exists a unique $¯ 𝜑 \in 𝖠 𝖻 (𝐺, 𝐻)$ such that $𝜑 = ¯ 𝜑 𝜋 𝐺$ , i.e. the following diagram commutes

Proof

$𝐺 a b$ is abelian by construction. By properties of the Kernel of a homomorphism into an abelian group, the universal property is equivalent to that of the quotient group.

This of course forms a Free-forgetful adjunction

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Table of Contents

Abelianization

Main theorem

Universal property

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