In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup ${\displaystyle N}$ of the group ${\displaystyle G}$ is normal in ${\displaystyle G}$ if and only if ${\displaystyle gng^{-1}\in N}$ for all ${\displaystyle g\in G}$ and ${\displaystyle n\in N.}$ The usual notation for this relation is ${\displaystyle N\triangleleft G.}$

Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of ${\displaystyle G}$ are precisely the kernels of group homomorphisms with domain ${\displaystyle G,}$ which means that they can be used to internally classify those homomorphisms.

Évariste Galois was the first to realize the importance of the existence of normal subgroups.

A subgroup ${\displaystyle N}$ of a group ${\displaystyle G}$ is called a normal subgroup of ${\displaystyle G}$ if it is invariant under conjugation; that is, the conjugation of an element of ${\displaystyle N}$ by an element of ${\displaystyle G}$ is always in ${\displaystyle N.}$ The usual notation for this relation is ${\displaystyle N\triangleleft G.}$

For any subgroup ${\displaystyle N}$ of ${\displaystyle G,}$ the following conditions are equivalent to ${\displaystyle N}$ being a normal subgroup of ${\displaystyle G.}$ Therefore, any one of them may be taken as the definition.

For any group ${\displaystyle G,}$ the trivial subgroup ${\displaystyle \{e\}}$ consisting of just the identity element of ${\displaystyle G}$ is always a normal subgroup of ${\displaystyle G.}$ Likewise, ${\displaystyle G}$ itself is always a normal subgroup of ${\displaystyle G.}$ (If these are the only normal subgroups, then ${\displaystyle G}$ is said to be simple.) Other named normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup ${\displaystyle [G,G].}$ More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup.

If ${\displaystyle G}$ is an abelian group then every subgroup ${\displaystyle N}$ of ${\displaystyle G}$ is normal, because ${\displaystyle gN=\{gn\}_{n\in N}=\{ng\}_{n\in N}=Ng.}$ More generally, for any group ${\displaystyle G}$, every subgroup of the center ${\displaystyle Z(G)}$ of ${\displaystyle G}$ is normal in ${\displaystyle G}$. (In the special case that ${\displaystyle G}$ is abelian, the center is all of ${\displaystyle G}$, hence the fact that all subgroups of an abelian group are normal.) A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.

A concrete example of a normal subgroup is the subgroup ${\displaystyle N=\{(1),(123),(132)\}}$ of the symmetric group ${\displaystyle S_{3},}$ consisting of the identity and both three-cycles. In particular, one can check that every coset of ${\displaystyle N}$ is either equal to ${\displaystyle N}$ itself or is equal to ${\displaystyle (12)N=\{(12),(23),(13)\}.}$ On the other hand, the subgroup ${\displaystyle H=\{(1),(12)\}}$ is not normal in ${\displaystyle S_{3}}$ since ${\displaystyle (123)H=\{(123),(13)\}\neq \{(123),(23)\}=H(123).}$ This illustrates the general fact that any subgroup ${\displaystyle H\leq G}$ of index two is normal.