Commutator Subgroup

The commutator subgroup (also called a derived group) of a group G is the subgroup generated by the commutators of its elements, and is commonly denoted G^' or [G,G]. It is the unique smallest normal subgroup of G such that G/[G,G] is Abelian (Rose 1994, p. 59). It can range from the identity subgroup (in the case of an Abelian group) to the whole group. Note that not every element of the commutator subgroup is necessarily a commutator.

For instance, in the quaternion group (+/-1, +/-i, +/-j, +/-k) with eight elements, the commutators form the subgroup (1,-1). The commutator subgroup of the symmetric group is the alternating group. The commutator subgroup of the alternating group A_n is the whole group A_n. When n>=5, A_n is a simple group and its only nontrivial normal subgroup is itself. Since [A_n,A_n] is a nontrivial normal subgroup, it must be A_n.

The first homology of a group G is the Abelianization


See also

Abelian Group, Abelianization, Cohomology Group, Commutator, Group, Normal Subgroup, Perfect Group

This entry contributed by Todd Rowland

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Rose, J. S. A Course on Group Theory. New York: Dover, 1994.

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Commutator Subgroup

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Rowland, Todd. "Commutator Subgroup." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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