Normal Subgroup

Definition

Let $G$ be a group and $N$ be a subgroup of $G$.

A subgroup $N$ is called a normal subgroup of $G$ if for every element $g \in G$,

$$gNg^{-1} = N$$

where

$$gNg^{-1} = \{ gng^{-1} \mid n \in N \}.$$

This means that conjugating any element of $N$ by any element of $G$ keeps the result inside $N$.

Equivalent ways to say that $N$ is normal in $G$ are:

• $gN = Ng$ for all $g \in G$
• Every left coset of $N$ is also a right coset
• $N$ is invariant under conjugation by elements of $G$

If $N$ is normal in $G$, we write:

$$N \trianglelefteq G$$

Main Content

1. Conjugation and Invariance

Conjugation is the key test for normality.

For a subgroup $N$ to be normal, every conjugate of every element of $N$ by every element of $G$ must still lie in $N$. In other words, if $n \in N$ and $g \in G$, then $gng^{-1} \in N$. This shows that $N$ is stable under the internal symmetry of the group.
Example: In the group $S_3$, the subgroup $A_3 = \{e, (123), (132)\}$ is normal because conjugating a 3-cycle by any permutation in $S_3$ gives another 3-cycle, and all 3-cycles are already in $A_3$.

Normal subgroups are structurally “compatible” with the whole group.

Ordinary subgroups may not remain fixed under conjugation, but normal subgroups do. This compatibility is why they allow the construction of a quotient group. Without normality, the product of cosets may not be well-defined, and the algebraic structure breaks down.
Example: In $S_3$, the subgroup $\{e, (12)\}$ is a subgroup but not normal, because conjugating $(12)$ by $(23)$ gives $(13)$, which is not in the subgroup.

2. Cosets and Quotient Groups

Normal subgroups make quotient groups possible.

If $N$ is normal in $G$, the set of left cosets $G/N = \{gN : g \in G\}$ can be given a group operation defined by $$(aN)(bN) = (ab)N.$$ This operation is well-defined only when $N$ is normal. The quotient group $G/N$ represents the “collapsed” version of $G$, where all elements of $N$ are treated as the identity-like part of the structure.

Left and right cosets coincide for normal subgroups.

For normal subgroups, $gN = Ng$ for every $g \in G$. This is a very useful practical criterion for checking normality. It means the subgroup does not depend on whether elements act from the left or the right.
Example: In an abelian group such as $(\mathbb{Z}, +)$, every subgroup is normal because $a + H = H + a$ automatically holds. For instance, $2\mathbb{Z}$ is normal in $\mathbb{Z}$.

3. Examples, Properties, and Tests

Examples of normal subgroups in common groups.

• In any abelian group, every subgroup is normal.
• In $S_3$, the subgroup $A_3$ is normal.
• In $\mathbb{Z}$, every subgroup $n\mathbb{Z}$ is normal.
• In any group $G$, the trivial subgroup $\{e\}$ and the whole group $G$ are always normal.
  These examples help show that normal subgroups occur naturally in many settings.

Important tests and properties.

A subgroup $N$ is normal if any one of the following holds:

• $gNg^{-1} = N$ for all $g \in G$
• $gN = Ng$ for all $g \in G$
• $N$ is the kernel of a group homomorphism
  The kernel property is especially important because every kernel is normal, and every normal subgroup can be realized as the kernel of some homomorphism. This connects normal subgroups to structure-preserving maps between groups.

Working / Process

1. Start with a subgroup $N$ of a group $G$.

First verify that $N$ is indeed a subgroup by checking closure, identity, and inverses. Without being a subgroup, it cannot be normal.

2. Test conjugation or cosets.

Check whether $gng^{-1} \in N$ for every $g \in G$ and $n \in N$. Alternatively, verify whether $gN = Ng$ for every $g \in G$. If either condition holds, $N$ is normal.

3. Use the normal subgroup to form a quotient group.

Once normality is confirmed, define the quotient group $G/N$ using cosets and the rule $(aN)(bN) = abN$. Then use the quotient to analyze the structure of $G$, simplify calculations, or study homomorphisms.

Advantages / Applications

Allows construction of quotient groups.

Normal subgroups are essential for building quotient groups, which are fundamental in understanding how large groups can be simplified into smaller factor groups.

Helps study homomorphisms and kernels.

Since every kernel of a group homomorphism is a normal subgroup, normal subgroups are directly connected to mappings between groups and the First Isomorphism Theorem.

Useful in analyzing symmetries and algebraic structure.

Normal subgroups help classify groups, study symmetry operations, and detect hidden structure in mathematical systems, including permutation groups, matrix groups, and symmetry groups in geometry and chemistry.

Summary

• A normal subgroup is a subgroup unchanged by conjugation from the group.
• Normality is the condition needed to form a quotient group.
• Every kernel of a homomorphism is a normal subgroup.
• Normal subgroups are central to understanding the internal structure of groups.