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).
$\blacksquare$