Normal Subgroup

A normal subgroup is a subgroup $N$ of a group $G$ that is preserved by conjugation by every element of $G$: $gNg^{-1}=N$ for all $g\in G$. Normal subgroups are central to group theory because they are exactly the subgroups that can be collapsed to the identity to produce a quotient group, and exactly the kernels of group homomorphisms. The notation $N\trianglelefteq G$, or $N\lhd G$, indicates that $N$ is normal in $G$. (jmilne.org)

Definition and equivalent conditions

For a subgroup $N\leq G$, the following conditions are equivalent:

• $gNg^{-1}=N$ for every $g\in G$.
• $gng^{-1}\in N$ whenever $g\in G$ and $n\in N$.
• The left and right cosets agree: $gN=Ng$ for every $g\in G$. (judsonbooks.org)

The operation $n\mapsto gng^{-1}$ is called conjugation. Normality requires preservation of the subgroup as a whole, not preservation of each element individually. Equivalently, a subgroup is normal precisely when it is a union of entire conjugacy classes of the ambient group. (ocw.mit.edu)

Normality is relative to the ambient group: a subgroup may be normal in one containing group but not in another. Every subgroup is normal in itself. (judsonbooks.org)

Quotient groups

For any subgroup $N$, the cosets form a set $G/N$. The proposed multiplication

defines a group operation independent of the chosen representatives exactly when $N$ is normal. In this quotient group, $N$ is the identity and $a^{-1}N$ is the inverse of $aN$. (jmilne.org)

Normality explains why representative choices do not matter. If $a'=an_1$ and $b'=bn_2$, with $n_1,n_2\in N$, then

Since $b^{-1}n_1b\in N$, this product lies in $abN$. Without normality, this reasoning can fail, so the coset set need not support the proposed multiplication. (jmilne.org)

Kernels and homomorphisms

For a group homomorphism $\varphi:G\to H$, its kernel

is normal, because

whenever $n\in\ker\varphi$. Conversely, every normal subgroup is the kernel of the canonical projection $G\to G/N$. (ocw.mit.edu)

The first isomorphism theorem states that

Thus, identifying elements that differ by an element of the kernel produces a group isomorphic to the image. (ocw.mit.edu)

Examples and non-examples

Abelian groups. Every subgroup of an abelian group is normal, since $gng^{-1}=n$. In the additive group of integers, $m\mathbb Z$ is normal, and $\mathbb Z/m\mathbb Z$, for $m\geq1$, describes addition modulo $m$. (judsonbooks.org)

Permutation groups. The alternating group $A_n$ is normal in the symmetric group $S_n$: it is the kernel of the sign homomorphism, which sends even permutations to $1$ and odd permutations to $-1$. For $n\geq2$, the quotient has two elements. (ocw.mit.edu)

For an explicit non-example, consider $H=\{e,(12)\}\leq S_3$. Conjugating $(12)$ by $(123)$ gives $(23)\notin H$, so $H$ is not normal. This follows directly from the conjugation criterion. (judsonbooks.org)

Matrix groups. Over a field $F$, the matrices of determinant $1$ form a normal subgroup $\mathrm{SL}_n(F)$ of $\mathrm{GL}_n(F)$, because they constitute the kernel of the determinant homomorphism. (jmilne.org)

Structural properties

Normal subgroups satisfy several useful closure and correspondence properties:

• An intersection of normal subgroups of $G$ is normal.
• If $N,M\trianglelefteq G$, their set product $NM$ is a normal subgroup.
• The inverse image of a normal subgroup under a homomorphism is normal.
• Under a surjective homomorphism, the image of a normal subgroup is normal.
• If $N\trianglelefteq G$, the correspondence $H\mapsto H/N$ matches subgroups of $G$ containing $N$ with subgroups of $G/N$, preserving normality in the respective ambient groups. (math.ucla.edu)

Related notions and applications

A characteristic subgroup is preserved by every automorphism of its group. This is stronger than normality, which requires preservation only by inner automorphisms—those arising from conjugation. Normality is not generally transitive: $K\trianglelefteq N$ and $N\trianglelefteq G$ need not imply $K\trianglelefteq G$. (jmilne.org)

A nontrivial group with no proper nontrivial normal subgroups is a simple group. Cyclic groups of prime order are simple, as are the alternating groups $A_n$ for $n\geq5$. Simplicity concerns normal subgroups, not the absence of proper subgroups altogether. (judsonbooks.org)

The commutator subgroup $[G,G]$, generated by elements $aba^{-1}b^{-1}$, is characteristic and therefore normal. It is the smallest normal subgroup whose quotient is abelian. The quotient $G/[G,G]$, called the abelianization, records the part of the group structure retained when commutativity is imposed. Repeated commutator subgroups form the derived series used to define a solvable group. (jmilne.org)

References

1. Group Theory — J. S. Milnejmilne.org
2. Abstract Algebra: Theory and Applications — Thomas W. Judsonjudsonbooks.org
3. Abstract Algebra — Chapter 2: Group Theory — Romyar Sharifimath.ucla.edu
4. RES.18-011 (Fall 2021) Full Lecture Notes: Algebra I Student Notesocw.mit.edu
5. Group Theory — J. S. Milne, version 3.15jmilne.org