Definition:Abelianization of Group

Definition

Let $G$ be a group.

The abelianization of $G$ is the quotient of $G$ by its derived subgroup:

$G^{\mathrm {ab} } = G / \sqbrk {G, G}$

Also see

• Derived Subgroup is Normal, ensuring the validity of this definition

• Abelianization of Group is Largest Abelian Quotient
• Universal Property of Abelianization of Group
• Definition:Abelianization of Group Functor

• Results about abelianizations of groups can be found here.

Linguistic Note

The British English form of abelianization is abelianisation.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ it is the norm to use American English, so the spelling abelianisation will rarely be encountered here, if at all.

Sources

• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Abelianization
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): commutator