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# Commutator Subgroup

The commutator subgroup (also called a derived group) of a group is the subgroup generated by the commutators of its elements, and is commonly denoted or . It is the unique smallest normal subgroup of such that 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 (, , , ) with eight elements, the commutators form the subgroup . The commutator subgroup of the symmetric group is the alternating group. The commutator subgroup of the alternating group is the whole group . When , is a simple group and its only nontrivial normal subgroup is itself. Since is a nontrivial normal subgroup, it must be .

The first homology of a group is the Abelianization

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

This entry contributed by Todd Rowland

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## References

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

## Referenced on Wolfram|Alpha

Commutator Subgroup

## Cite this as:

Rowland, Todd. "Commutator Subgroup." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CommutatorSubgroup.html