Equivalence of Definitions of Normal Subset/1 iff 2

Theorem
Let $\struct {G, \circ}$ be a group.

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

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

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

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

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

Let $g \in G$.

Then:

Sufficient Condition
Suppose that $S$ satisfies $(2)$ or $(3)$.

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

Let $g \in G$.

Then: