Commutator subgroup
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In mathematics, the commutator subgroup (or derived subgroup) of a group G is the subgroup <math>[G,G]</math> (also denoted <math>G'</math> or <math>G^{(1)}</math>) generated by all the commutators of elements of G; that is, <math>[G,G]</math> = <[g,h] : g,h in G>.
If <math>[G,G]=G</math>, G is called perfect group.
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Examples
- An abelian group has only trivial commutators. Hence its commutator subgroup is {1}. The converse is also (trivially) true.
- The commutator of the symmetric group <math>S_n</math> is the alternating group <math>A_n</math>
Notes
The commutator subgroup can also be defined as the set of elements g of the group which have an expression as a product g=g1g2...gk that can be rearranged to give the identity.
Note that the set of all commutators of the group is, generally, not a group (in any interesting case). While clumsily defined, the commutator subgroup is important.
The derived subgroup, in a sense, gives a measure of how far G is from being abelian; the larger <math>[G,G]</math>, the "less abelian" G is. In particular, <math>[G,G]</math> is equal to {1} if and only if the group G is abelian.
Properties
If f : G → H is a group homomorphism, then <math>f([G,G])</math> is a subgroup of <math>[H,H]</math>, because f maps commutators to commutators. This implies that the operation of forming derived groups is a functor from the category of groups to the category of groups.
Applying this to endomorphisms of G, we find that <math>[G,G]</math> is a fully characteristic subgroup of G, and in particular a normal subgroup of G. (To reach the final conclusion, simply take conjugation with any particular g in G to be the automorphism in question. We see that g-1<math>[G,G]</math>g = <math>[G,G]</math> for every g in G, and therefore that <math>[G,G]</math> is a normal subgroup of G.). The quotient <math>G/[G,G]</math> is an abelian group sometimes called G made abelian, or the abelianization of G. In a sense, it is the abelian group that's "closest" to G, which can be expressed by the following universal property: if p : G → <math>G/[G,G]</math> is the canonical projection, and f : G → A is any homomorphism from G to an abelian group A, then there exists exactly one homomorphism s : <math>G/[G,G]</math> → A such that s o p = f. In the language of category theory: the functor which assigns to every group its abelianization is left adjoint to the forgetful functor which assigns to every abelian group its underlying group.
In particular, a quotient G/N of G is abelian if and only if N includes <math>[G,G]</math>.