Equivalence of Definitions of Normal Subset/3 iff 5

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

Let $S \subseteq G$.

Then:
 * $S$ is a normal subset of $G$ by Definition 3


 * $S$ is a normal subset of $G$ by Definition 5.
 * $S$ is a normal subset of $G$ by Definition 5.

3 implies 5
Suppose that $S$ is a normal subset of $G$ by Definition 3.

That is:


 * $\forall g \in G: g^{-1} \circ S \circ g \subseteq S$.

Let $x, y \in G$ such that $x \circ y \in S$.

Then:

5 implies 3
Suppose that $S$ is a normal subset of $G$ by Definition 5.

That is:
 * $\forall x, y \in G: x \circ y \in S \implies y \circ x \in S$

Let $g \in G$.

Then: