A normal subgroup H of a group G is a subgroup of G that is invariant under conjugation by members of the group. In other words, every left coset and right coset corresponding to an element g are the same, that is, gH=Hg. Normal subgroups have many applications. In this post, we will learn about normal subgroups with examples, their properties and related theorems.

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## Definition of Normal Subgroup

A subgroup H of a group G is called a normal subgroup of G if H is invariant under conjugation by any element of G. That is,

gHg^{-1} = H ∀ g ∈ G.

**Notation:** If H is a normal subgroup of G, then we denote it by H◁G.

For example, every subgroup of index two is normal.

Normal subgroups are also known as invariant subgroups or distinguished subgroups or self-conjugate subgroups.

### Equivalent Definitions of Normal Subgroups

The following are a few equivalent conditions that also define a normal subgroup. For a subgroup H of G, we say H is normal in G if one of the following is satisfied.

- gHg
^{-1}⊆ H ∀ g ∈ G. That is, the image of conjugation of H by any element of G lies inside H. - For any element g ∈ G, the left coset gH and the right coset are equal, i.e., gH=Hg ∀ g ∈ G.
- The set of all left cosets of G = The set of all right cosets of G.
- The union of conjugacy classes of G is H.
- There exists a homomorphism G → G’ with kernel H.

## Examples of Normal Subgroups

The trivial subgroup {e_{G}} and the improper subgroup G of a group G are always normal in G. Other than these subgroups, below are a few examples of normal subgroups.

- The alternating group A
_{3}is a normal subgroup of S_{3}. This is because the index [S_{3}: A_{3}] = 2 and we know that subgroups of index 2 are normal. - The centre of a group G is defined as Z(G) = {g∈ G : gx=xh ∀ x ∈ G}. The centre Z(G) of any group G is a normal subgroup of G.

## Normal Subgroup Test

Let H be a subgroup of a group G. Then H is normal in G if and only

ghg^{-1} ∈ H, for all g ∈ G.

In other words, we say that H◁G if and only if the conjugate of an element of H by any element of G belongs to H.

Also Read:Order of a Group: The order of a group and of its elements are discussed here with formulas. Abelian Group: Definition, Properties, Examples Cyclic Group: The definition, properties, and related theorems on cyclic groups are discussed.Left Coset and Right Coset: Definition, Examples, Properties and TheoremsFirst Isomorphism Theorems of GroupsKernel of a Group Homomorphism |

## Properties of Normal Subgroups

- If H◁G and K is a subgroup of G such that H ⊆ K, then H◁K.
- Let K be a normal subgroup of G and H be a normal subgroup of K such that H ⊂ K ⊂ G. Then H need not be a normal subgroup of G.
- Both G and H are normal subgroups of their direct product G×H.
- If H is an index 2 subgroup of G, then H is a normal subgroup of G.
- If H
_{1}and H_{2}are two normal subgroups of G_{1}and G_{2}. Then the direct product H_{1}×H_{2}is a normal subgroup of G_{1}×G_{2}.

## Normal Subgroup Theorems

**Theorem 1:** Every subgroup of an abelian group is normal.

*Proof:*

**Theorem 2:** Let H be a subgroup of G and [G : H] = 2. Then H is a normal subgroup of G.

*Proof:*