A normal subgroup H of a group G is a subgroup of G which satisfies the similarity transformation with any fixed arbitrary element in G. if G is an abelian group and x is an arbitrary element of G, then Hx is a right coset of H in G and xH is a left coset of in G. Since G is abelian then xH = Hx. However, it is possible that even though G is not an abelian group, it satisfies xH = Hx. In this case subgroups of G are said to be normal subgroups or invariant subgroups or distinguished subgroups.

Definition of Normal Subgroup

Let H be a subgroup of G, then H is said to be a normal subgroup of G, if for every x in G and for h in H xh = xh, that is, xhx^-1 belongs to H.

Now since the above statement is true for all h in H. Therefore, we can have

xHx^-1 = {xhx^-1: for all h in H}, thus normal subgroups of a group G can be defined as:

Normal subgroups are sometimes also referred to as self-conjugates. Normal subgroups are denoted as H ◁ G, it is read as “H is a normal subgroup of G”.

Examples of Normal Subgroup

Every group has necessarily two trivial normal subgroups, viz., the single identity element of G and G itself.

• Let e be the identity element in G, then {e} will be a trivial subgroup of G. Now for every g in G, there exist g^-1 in G, then

geg^-1 = gg^-1 = e ∈ {e}

Thus {e} is the normal subgroup of G.

• Since G is closed with respect to multiplication of its elements, let g, h be any two elements of G, then

ghg^-1 = k which could be any element in G. Thus, G itself is a normal subgroup.

Another example of a normal subgroup could be the centre of the group. Let Z be the centre of the group G such that

Z(G) = { z ∈ G | for every g in G, gz = zg}, that elements of Z commute with every element of G.

Clearly, gzg^-1 = gg^-1z = ez = z ∈ Z; where e is an identity element in G.

Thus, Z is a normal subgroup of G.

Properties of a Normal Subgroup

• The intersection of any two normal subgroups of a group is a normal subgroup.
• The intersection of any collection of normal subgroups is a normal subgroup. That is,

If N₁, N₂, …., N_r are normal subgroups of a group G, then N₁ ∩ N₂ ∩, ….,∩ N_r is also a normal subgroup of G.

• Every abelian group has a normal subgroup.
• Any group which do not have any normal subgroup other than the trivial normal subgroup is called a simple group.
• If a subgroup is of index 2 in G, that is has only two distinct left or right cosets in G, then H is a normal subgroup of G.
• Every subgroup of a cyclic group is normal.

Related Articles:

• Group
• Linear Algebra
• Group Theory
• Abstract Algebra

Solved Examples on Normal Subgroup

Example 1:

Prove that a subgroup H of a group G is a normal subgroup if and only if g^-1Hg = H for every g in G.

Solution:

We first take for H be a subgroup of H, g^-1Hg = H for every g in G. Then

gHg^-1 = H for every g in G

⇒ gHg^-1 ⊆ H for every g in G

Hence, by the definition of normal subgroup H is a normal subgroup of G.

Conversely, let H be a normal subgroup of G, then

gHg^-1 ⊆ H for every g in G

⇒ H ⊆ g^-1Hg for every g in G ………….(1)

Now, H ⊆ (g^-1)^-1Hg^-1 for every g in G

⇒ H ⊆ gHg^-1 for every g in G

⇒ g^-1 H g ⊆ g^-1(gHg^-1)g for every g in G

⇒ g^-1 H g ⊆ (g^-1g)H(g^-1g) for every g in G

⇒ g^-1 H g ⊆ eHe; e is identity in G

⇒ g^-1 H g ⊆ H for every g in G ………..(2)

From (1) and (2), if H is a normal subgroup of G then g^-1Hg = H for every g in G.

Example 2:

Prove that all abelian groups have normal subgroups.

Solution:

Let G be an abelian group and H be a subgroup of G. Since G is abelian therefore all elements of G commutative with respect to multiplication. Let g ∈ G and h ∈ H as H is a subgroup of G the h must belong to G also, then

hg = gh for all g in G

⇒ ghg^-1 = h ∈ H for all g in G

⇒ gHg^-1⊆ H for all g in G

Hence, every abelian group has normal subgroups.