# Which subgroups are normal?

## Which subgroups are 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. Equivalently, a subgroup H of G is normal if and only if g H = H g gH = Hg gH=Hg for any g ∈ G g \in G g∈G.

### How do you find the normal subgroups of a group?

Let G be a group and S < G such that [G : S] = 2: Then S is a normal subgroup of G. Since An is a subgroup of order n!/2 and index 2 in Sn. Therefore An is a normal subgroup of Sn. Theorem.

What are the normal subgroups of S3?

There are three normal subgroups: the trivial subgroup, the whole group, and A3 in S3.

Is a subgroup of G?

A subset H of the group G is a subgroup of G if and only if it is nonempty and closed under products and inverses. The identity of a subgroup is the identity of the group: if G is a group with identity eG, and H is a subgroup of G with identity eH, then eH = eG.

## How do you prove a subgroup is normal?

The best way to try proving that a subgroup is normal is to show that it satisfies one of the standard equivalent definitions of normality.

1. Construct a homomorphism having it as kernel.
2. Verify invariance under inner automorphisms.
3. Determine its left and right cosets.
4. Compute its commutator with the whole group.

### Is S2 a normal subgroup of S3?

S2 is not normal in S3.

Why is every subgroup of index 2 normal?

Theorem: A subgroup of index 2 is always normal. Proof: Suppose H is a subgroup of G of index 2. Then there are only two cosets of G relative to H . Then G can be decomposed into the cosets H,sH H , s H or H,Hs H , H s , implying H commutes with s .

What are the normal subgroups of A4?

The group A4 has order 12, so its subgroups could have size 1, 2, 3, 4, 6, or 12. There are subgroups of orders 1, 2, 3, 4, and 12, but A4 has no subgroup of order 6 (equivalently, no subgroup of index 2). Here is one proof, using left cosets. Theorem 1.

## Is any subgroup of a normal subgroup normal?

A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8. However, a characteristic subgroup of a normal subgroup is normal.