What_is_the_commutator_subgroup@20250822_200544

user: What is the commutator subgroup? [Timestamp: 2025/08/22 20:05:44]

ai: The commutator subgroup of a group is defined as the subgroup generated by all the commutators of the group. To be precise, for any two elements $a,b\in G$, the commutator is defined as

$$[a,b]=ab{a}^{-1}{b}^{-1}.$$

Then the commutator subgroup is given by

$$[G,G]=⟨[a,b]:a,b\in G⟩.$$

This subgroup has some important properties:

• It is a normal subgroup of $G$.
• The quotient $G/[G,G]$ is an abelian group, meaning that $[G,G]$ is the smallest normal subgroup of $G$ such that the quotient is abelian.
• If two elements in $G$ do not commute, their commutator is not the identity.

As quoted in the source:

Commutator and Commutator Subgroup

The commutator of two elements $a$ and $b$ in a group G is the element $[a,b]=ab{a}^{-1}{b}^{-1}$. The commutator subgroup of $G$ is the group generated by all commutators: $[G,G]:=⟨[a,b]:a,b\in G⟩$
Prop $[G,G]$ is a normal subgroup of $G$.
Prop $G_{\text{ab}}:=G/[G,G]$ is abelian.

This provides the complete picture of what the commutator subgroup is.

Sources:

• Subgroups Generated by Subsets
• Groups, Order and Subgroups
• Cyclic Groups [Timestamp: 2025/08/22 20:05:56]