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Given a group $G$, the commutators of $G$ are the elements $a^{-1}b^{-1}ab$ where $a,b\in G$.

The subgroup generated by all commutators of $G$ is the commutator subgroup or derived subgroup of $G$, and is denoted by $[G,G]$ or $G'$.

The commutator subgroup is always a normal subgroup, and the quotient group $G/G'$, called the abelianization of $G$, is the largest abelian quotient of $G$ in the sense that any homomorphism $G\to A$ where $A$ is an abelian group factors as a composition of homomorphisms $G\to G/G'\to A$.

Knowl status:
  • Review status: reviewed
  • Last edited by Sam Schiavone on 2021-07-12 19:20:33
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