Definition:Normal Subgroup/Definition 5

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Definition

Let $G$ be a group.

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


$N$ is a normal subgroup of $G$ if and only if:

\(\ds \forall g \in G: \, \) \(\ds N\) \(=\) \(\ds g \circ N \circ g^{-1}\)
\(\ds \forall g \in G: \, \) \(\ds N\) \(=\) \(\ds g^{-1} \circ N \circ g\)

where $g \circ N$ etc. denotes the subset product of $g$ with $N$.


That is:

$\forall g \in G: \map {\kappa_g} N = N$

where $\map {\kappa_g} N$ denotes the inner automorphism of $N$ by $g$.


Notation

The statement that $N$ is a normal subgroup of $G$ is represented symbolically as $N \lhd G$.


A normal subgroup is often represented by the letter $N$, as opposed to $H$ (which is used for a general subgroup which may or may not be normal).


To use the notation introduced in the definition of the conjugate:

$N \lhd G \iff \forall g \in G: N^g = N$


Also known as

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:

\(\ds \forall g \in G: \, \) \(\ds \leftparen {n \in N}\) \(\iff\) \(\ds \rightparen {g \circ n \circ g^{-1} \in N}\)
\(\ds \forall g \in G: \, \) \(\ds \leftparen {n \in N}\) \(\iff\) \(\ds \rightparen {g^{-1} \circ n \circ g \in N}\)


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.


Also see

  • Results about normal subgroups can be found here.


Sources

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