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Normal Subgroup — Definition, Formula & Examples

A normal subgroup is a subgroup that remains unchanged when every element is conjugated by any element of the larger group. This special property is exactly what you need to form a well-defined quotient group.

A subgroup NN of a group GG is called normal in GG, written N⊴GN \trianglelefteq G, if for every g∈Gg \in G, the conjugate gNg−1=NgNg^{-1} = N. Equivalently, NN is normal if and only if every left coset of NN equals the corresponding right coset: gN=NggN = Ng for all g∈Gg \in G.

Key Formula

N⊴G  ⟺  gNg−1=Nfor all g∈GN \trianglelefteq G \iff gNg^{-1} = N \quad \text{for all } g \in G
Where:
  • NN = A subgroup of the group G
  • GG = The ambient group
  • gNg−1gNg^{-1} = The set $\{gng^{-1} : n \in N\}$, called the conjugate of N by g

How It Works

To check whether a subgroup NN is normal in GG, pick an arbitrary element g∈Gg \in G and an arbitrary element n∈Nn \in N, then verify that gng−1∈Ngng^{-1} \in N. If this holds for every such pair, NN is normal. A useful shortcut: every subgroup of an abelian group is automatically normal, since gng−1=gg−1n=n∈Ngng^{-1} = gg^{-1}n = n \in N. Also, any subgroup of index 2 is always normal, because there is only one left coset besides NN itself, forcing left and right cosets to coincide.

Worked Example

Problem: Let G=S3G = S_3 (the symmetric group on 3 elements) and let N={e,(1 2 3),(1 3 2)}N = \{e, (1\,2\,3), (1\,3\,2)\}. Show that NN is a normal subgroup of S3S_3.
Identify the subgroup and its index: N consists of the identity and the two 3-cycles. Since ∣S3∣=6|S_3| = 6 and ∣N∣=3|N| = 3, the index is [S3:N]=2[S_3 : N] = 2.
[S3:N]=∣S3∣∣N∣=63=2[S_3 : N] = \frac{|S_3|}{|N|} = \frac{6}{3} = 2
Apply the index-2 criterion: Any subgroup of index 2 is normal. There are only two cosets: NN itself and the single remaining coset. So left cosets and right cosets must be the same.
gN=Ngfor all g∈S3gN = Ng \quad \text{for all } g \in S_3
Conclude: Since gN=NggN = Ng for every g∈S3g \in S_3, the normality condition is satisfied.
N⊴S3N \trianglelefteq S_3
Answer: N={e,(1 2 3),(1 3 2)}N = \{e, (1\,2\,3), (1\,3\,2)\} is a normal subgroup of S3S_3, which also yields the quotient group S3/N≅Z2S_3 / N \cong \mathbb{Z}_2.

Why It Matters

Normal subgroups are the building blocks of quotient groups, which appear throughout algebra, number theory, and physics. The First Isomorphism Theorem, a cornerstone of any abstract algebra course, requires the kernel of a homomorphism to be a normal subgroup. Understanding normality is essential for classifying finite groups and for applications in cryptography and coding theory.

Common Mistakes

Mistake: Assuming gng−1=ngng^{-1} = n (element-wise fixing) is required for normality.
Correction: Normality only requires that gng−1gng^{-1} lands somewhere in NN, not that each element is individually fixed. The conjugate of nn can be a different element of NN.