Commutator subgroup

In mathematics, the commutator group called (or commutator subgroup ) of a group that subgroup generated by the commutators in the group:

The commutator is also referred to.

In general, the set of all commutators is not a subset of, the phrase generated by the definition so it can not be omitted.

The inverse of a commutator is given by, corresponding to products of commutators.

The order of the commutator gives an indication of how far a group is removed from the commutativity. A group is commutative if and only ( Abelian ) if their commutator group consists only of the identity element. In this case applies namely. Groups in which the commutator subgroup, however, covers the whole group, hot perfect groups.

Properties

Since the amount of the commutators is mapped under each automorphism of to be the commutator subgroup is a characteristic subgroup of and hence a normal subgroup of the group.

The factor group is always abelian, it is called Abelisierung the group. For every normal subgroup applies:

That is, the commutator subgroup is the smallest normal subgroup for which the quotient group is abelian.

Example

It is the symmetric group and the alternating group. Then:

• , Said Klein's four group means.

Higher Kommutatorgruppen

Forming the commutator can be iterated, refers to the -th commutator with. The recursive definition is:

A group is called solvable if and only if a descending chain of Subnormalteilern ( Subnormalreihe ) exists, so that the factor groups are abelian. The construction of the iterated commutator provides a criterion for the dissolubility of:

Either the resulting on continued Kommutatorbildung descending series of subgroups or a refinement of this series are equivalent to each such Subnormalreihe or a refinement thereof.

The relationship between the two equivalent definitions of resolvability, on continued Kommutatorenbildung one hand and a Subnormalreihe other hand, and the concept of Subnormalreihe themselves are detailed in the article " series ( group theory ) " below.

Example

The symmetric group or the alternating group is then exactly solvable when. For you see the immediately with the above example. For the following applies:

For the chain of iterated Kommutatorgruppen is in stationary, then, is not yet resolved.