DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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The abelianisation $G^{a b} : = G / [G, G]$ and its canonical map

Definition

Let $G$ be a group. Its abelianisation is the quotient

$G^{a b} : = G / [G, G],$

where $[G, G]$ is the commutator subgroup of Commutators $[g, h] = g h g^{- 1} h^{- 1}$ and the commutator subgroup $[G, G]$ . This subgroup is normal by The commutator subgroup is normal, so the quotient is defined. The abelianisation map is the canonical surjective homomorphism

$q_{G} : G ⟶ G^{a b}, q_{G} (g) = g [G, G],$

of The canonical projection $π : G \to G / N$ , $π (g) = g N$ , is a surjective group homomorphism.

Depends on

Used by

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John McKernan, Presentations and Groups of Small Order, Lecture 12 (standard reference, not scraped)
Encyclopedia of Mathematics, Commutator subgroup (standard reference, not scraped)

The abelianisation Gab:=G/[G,G]Gab:=G/[G,G] and its canonical map