Equivalence of Definitions of Normal Subset/1 iff 2

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

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

Then Normal Subset/Definition 1 is equivalent to Normal Subset/Definition 2. That is, the following three statements are equivalent:


 * $(1)\quad \forall g \in G: g \circ N = N \circ g$


 * $(2)\quad \forall g \in G: g \circ N \circ g^{-1} = N$
 * $(3)\quad \forall g \in G: g^{-1} \circ N \circ g = N$

Proof
Let $e$ be the identity of $G$.

First note that:
 * $(4): \quad \left({\forall g \in G: g \circ N \circ g^{-1} = N}\right) \iff \left({\forall g \in G: g^{-1} \circ N \circ g = N}\right)$

which is shown by, for example, setting $h := g^{-1}$ and substituting.

Necessary Condition
Suppose that $N$ satisfies $(1)$.

Then:

Sufficient Condition
Let $N$ be a subset of $G$ such that:
 * $\forall g \in G: g \circ N \circ g^{-1} = N$ or
 * $\forall g \in G: g^{-1} \circ N \circ g = N$

By $(4)$, as long as one of these statements holds, the other one holds as well.

Then: