Equivalence of Definitions of Normal Subset/2 implies 3

Theorem

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

Let $S \subseteq G$.

Let $S$ be a normal subset of $G$ by Definition 2.

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

That is, if:

$\forall g \in G: g \circ S \circ g^{-1} = S$

or:

$\forall g \in G: g^{-1} \circ S \circ g = S$

then:

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

and:

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

Proof

We have that:

$\left({\forall g \in G: g \circ S \circ g^{-1} = S}\right) \iff \left({\forall g \in G: g^{-1} \circ S \circ g = S}\right)$

The result follows by definition of set equality.

$\blacksquare$