Subgroup is Normal iff Normal Subset

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Theorem

Let $\left({G, \circ}\right)$ be a group.

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


Then $N$ is normal in $G$ (by definition 1) if and only if it is a normal subset of $G$.


Proof

Necessary Condition

Let $N$ be normal in $G$ (by definition 1):

Thus for each $g \in G$:

$\forall g \in G: g \circ N = N \circ g$

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


Thus $N$ is a normal subset of $G$ (by definition 1):

$\forall g \in G: g \circ N = N \circ g$


Sufficient Condition

Let $N$ be a normal subset of $G$ (by definition 1):

$\forall g \in G: g \circ N = N \circ g$

Since $N$ is a subgroup, $N$ is a normal subgroup of $G$ (by definition 1).

$\blacksquare$