Definition:Normal Subgroup/Also known as

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It is usual to describe a normal subgroup of $G$ as normal in $G$.

Some sources refer to a normal subgroup as an invariant subgroup or a self-conjugate subgroup.

This arises from Definition 6:

$\forall g \in G: \paren {n \in N \iff g \circ n \circ g^{-1} \in N}$
$\forall g \in G: \paren {n \in N \iff g^{-1} \circ n \circ g \in N}$

which is another way of stating that $N$ is normal if and only if $N$ stays the same under all inner automorphisms of $G$.

See Inner Automorphism Maps Subgroup to Itself iff Normal.

Some sources use distinguished subgroup.