Normal Subgroup: Definition, Examples, Properties, Theorems

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.

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.

1. The alternating group A₃ is a normal subgroup of S₃. This is because the index [S₃ : A₃] = 2 and we know that subgroups of index 2 are normal.
2. 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 Theorems First Isomorphism Theorems of Groups Kernel of a Group Homomorphism

Properties of Normal Subgroups

1. If H◁G and K is a subgroup of G such that H ⊆ K, then H◁K.
2. 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.
3. Both G and H are normal subgroups of their direct product G×H.
4. If H is an index 2 subgroup of G, then H is a normal subgroup of G.
5. If H₁ and H₂ are two normal subgroups of G₁ and G₂. Then the direct product H₁×H₂ is a normal subgroup of G₁×G₂.

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: