# What is the commutator subgroup of a group?

## What is the commutator subgroup of a group?

The commutator subgroup is the group generated by all products ghg-1h-1 for g and h ∈ G. It is denoted [G, G]. The commutator subgroup is also called the derived subgroup, and is sometimes also denoted G/ or D(G).

What is a commutator of a group?

The commutator of two elements, g and h, of a group G, is the element. [g, h] = g−1h−1gh. This element is equal to the group’s identity if and only if g and h commute (from the definition gh = hg [g, h], being [g, h] equal to the identity if and only if gh = hg).

### How do you find the commutator subgroup of a group?

The commutator subgroup can also be defined as the set of elements g of the group that have an expression as a product g = g1 g2 gk that can be rearranged to give the identity.

What is commutator subgroup of S3?

The commutator subgroup of S3 is A3, so the commutator subgroup of Z × S3 is {(0,x)|x is an even permutation}. (b) Calculate the factor group of Z × S3 over its commutator subgroup. The factor group of S3 over A3 is Z2, so the factor group of Z × S3 over its commutator subgroup is Z × Z2.

## Is the commutator a normal subgroup?

Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup Let G be a group and H and K be subgroups of G. For h∈H, and k∈K, we define the commutator [h,k]:=hkh−1k−1. Let [H,K] be a subgroup of G generated by all such commutators.

How do you find the commutator group?

The subgroup C of G is called the commutator subgroup of G, and it general, it is also denoted by C = G or C = [G, G], and is also called the derived subgroup of G. If G is Abelian, then we have C = {e}, so in one sense the commutator subgroup may be used as one measure of how far a group is from being Abelian.

### What is the function of commutator?

The commutator on the DC generator converts the AC into pulsating DC. The commutator assures that the current from the generator always flows in one direction. The brushes ride on the commutator and make good electrical connections between the generator and the load.

What does the commutator tell you?

A commutator in quantum mechanics tells us if we can measure two ‘observables’ at the same time. So the commutator tells us that observing the system in different orders effects the outcome. Going back to the Heisenberg example, taking a measurement of the position of a particle will disturb its momentum.

## Is the commutator subgroup abelian?

The groups generated by S and T satisfying the relations S3 = T2=(ST)i = 1 were classified by Professor Miller, f The fact that makes these groups particularly easy to manage is that the commutator subgroups are abelian. generated by two operators of order two are the well known dihedral groups.

What is the commutator subgroup of S4?

Since S4/A4 is abelian, the derived subgroup of S4 is con- tained in A4. Also (12)(13)(12)(13) = (123), so that (nor- mality!) every 3-cycle is a commutator.

### Is normal subgroup abelian?

A subgroup of a group is termed an abelian normal subgroup if it is abelian as a group and normal as a subgroup.

For what condition a subgroup is normal?

A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group: H is normal if and only if g H g − 1 = H gHg^{-1} = H gHg−1=H for any. g \in G. g∈G.

## Which is the group generated by a commutator subgroup?

Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup Let G be a group and H and K be subgroups of G. For h ∈ H, and k ∈ K, we define the commutator [ h, k] := h k h − 1 k − 1 .

Which is the abelian quotient group generated by commutators?

Here [ H, G] is a subgroup of G generated by commutators [ h, k] := h k h − 1 k − 1 . In particular, the commutator subgroup [ G, G] is a normal subgroup of […] Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup Let G be a group and H and K be subgroups of G.

### Which is a fully characteristic subgroup of G?

The commutator subgroup of G is usually denoted by [ G, G], G ′ or Γ 2 ( G). The commutator subgroup is a fully-characteristic subgroup, and any subgroup containing the commutator subgroup is a normal subgroup.

Which is the smallest subgroup of a group?

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. 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. In other words, .