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What is a commutator subgroup

  1. Jul 23, 2014 #1
    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:
    [itex][g,h] = g h g^{-1} h^{-1}[/itex]

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

    Extended explanation

    One can extend the concept of commutator subgroup further.

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

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

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

    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.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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