Subgroup is Normal iff Normal Subset

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) iff it is a normal subset of $G$.

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

Thus for each $g \in G$:
 * $\forall g \in G: g \circ N = N = N \circ g$

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

Then by applying the subset product of $g^{-1}$ with $g \circ N$:

Thus $N$ is a normal subgroup of $G$ (by definition 5).