commutator subgroup

In a sense, N can be thought of as the closure of the commutator subgroup.

The higher commutator subgroups of a group, Bull.

The commutator subgroup of a group is always a fully characteristic subgroup.

Examples of characteristic subgroups include the commutator subgroup and the center of a group.

These correspond to the center and the commutator subgroup (for upper and lower central series, respectively).

The commutator subgroup of the alternating group A is the Klein four group.

The binary icosahedral group is perfect, meaning that it is equal to its commutator subgroup.

The commutator subgroup is a normal subgroup.

This shows that the commutator subgroup can be viewed as a functor on the category of groups, some implications of which are explored below.

In mathematics, a metabelian group is a group whose commutator subgroup is abelian.

In other words, G/N is abelian if and only if N contains the commutator subgroup.

An IA automorphism is thus an automorphism that sends each coset of the commutator subgroup to itself.

If H is the commutator subgroup G' of G, then the corresponding transfer map is trivial.

All Abelian groups are isoclinic since they are equal to their centers and their commutator subgroups are always the identity subgroup.

As the elementary matrices generate the commutator subgroup, this map is onto the commutator subgroup.

The commutator subgroup of is isomorphic to the commutator subgroup of .

The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian.

So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.

The symmetric group S of order 24 is solvable but is not metabelian because its commutator subgroup is the alternating group A which is not abelian.

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

Since the Alexander ideal is principal, if and only if the commutator subgroup of the knot group is perfect (i.e. equal to its own commutator subgroup).

In algebraic K-theory, a field of mathematics, the Steinberg group St(A) of a ring A, is the universal central extension of the commutator subgroup of the stable general linear group.

The smallest (non-trivial) perfect group is the alternating group A. More generally, any non-abelian simple group is perfect since the commutator subgroup is a normal subgroup with abelian quotient.

In mathematics, in the realm of group theory, a Group (mathematics) is said to be 'perfect' if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian group quotient group.

Two related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the commutator subgroup of GL, and the group generated by transvections.