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normal subgroup

A subgroup $N$ of a group $G$ is normal if the conjugation $n\mapsto g^{-1}ng$ by any element $g\in G$ leaves $N$ invariant, i.e. $g^{-1}Ng:=\{g^{-1}ng\mid n\in N\}=N$.

A subgroup $N$ is normal iff the partition of the group into left cosets of the subgroup $N$, that is the sets $gN=\{gn\mid n\in N\}$, is stable in the sense that the left coset $g_1{g}_{2}N$ of the product $g_1{g}_2$ of any two elements $g_1,g_2\in G$ depends only on the coset $g_{1}N$, $g_{2}N$. Thus there is well defined product on the set of cosets making the set of left cosets $N\backslash G$ a group. By $gN=gN{g}^{-1}g=Ng$ the set of left cosets and the set of right cosets of a normal subgroup coincide; thus the induced group structure on the right coset set $G/N$ is the same and called the quotient group? (see quotient object).

A normal subgroup is a normal subobject of a group in the category of groups: the more general notion of ‘normal subobject’ makes sense in semiabelian categories and some other setups. If we consider a group as a special case of an $\Omega$-group, then a normal subgroup corresponds to an ideal?.

Of course, every subgroup of an abelian group is normal.