What is a Commutator Subgroup?

[Greg Bernhardt]

Definition/Summary

The commutator subgroup of a group is the subgroup generated by commutators of all the elements. For group G, it is [G,G]. Its quotient group is the maximal abelian quotient group of G.

Equations

The commutator of group elements g, h:
$[g,h] = g h g^{-1} h^{-1}$

The commutator of groups G and H is
$[G,H] = \text{group generated by} \{[g,h] : g \in G ,\ h \in H\}$

Extended explanation

One can extend the concept of commutator subgroup further.

The derived series of a group G is
$G^{(0)} = G$
$G^{(n)} = [G^{(n-1)}, G^{(n-1)}]$

If it converges on the identity group, then G is solvable.

The lower central series of a group G is
$G_0 = G$
$G_n = [G_{n-1}, G]$

If it converges on the identity group, then G is nilpotent.

Every nilpotent group is solvable, though the converse is not necessarily true. There are some solvable groups that are not nilpotent, and the smallest one of these is the smallest nonabelian group, the symmetric group S(3) ~ dihedral group D(3).

The quotient groups between successive members of both series are always abelian.

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[fresh_42]

Btw., it is not by chance that Lie algebra multiplication is written as $[A,B]$. If we have a Lie group $G$ with Lie algebra $\mathfrak{g}$, then $[G,G]=\{\,ghg^{-1}h^{-1}\,\}$ is the Lie group of $[\mathfrak{g},\mathfrak{g}]=\{\,[A,B]\,\}$ (up to technical details).