Equivalence of Definitions of Normal Subset/3 iff 4

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 4.
 * $S$ is a normal subset of $G$ by Definition 4.

That is, the following conditions are equivalent:
 * $(1) \quad \forall g \in G: g \circ S \circ g^{-1} \subseteq S$
 * $(2) \quad \forall g \in G: g^{-1} \circ S \circ g \subseteq S$
 * $(3) \quad \forall g \in G: S \subseteq g \circ S \circ g^{-1}$
 * $(4) \quad \forall g \in G: S \subseteq g^{-1} \circ S \circ g$

Proof
First note that:


 * $(5): \quad \paren {\forall g \in G: g \circ S \circ g^{-1} \subseteq S} \iff \paren {\forall g \in G: g^{-1} \circ S \circ g \subseteq S}$


 * $(6): \quad \paren {\forall g \in G: S \subseteq g \circ S \circ g^{-1}} \iff \paren {\forall g \in G: S \subseteq g^{-1} \circ S \circ g}$

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

Therefore:
 * conditions $(1)$ and $(2)$ are equivalent

and:
 * conditions $(3)$ and $(4)$ are equivalent.

It remains to be shown that condition $(1)$ is equivalent to condition $(3)$.

Suppose that $(1)$ holds.

Then:

Thus condition $(1)$ implies condition $(3)$.

The exact same argument, substituting $\supseteq$ for $\subseteq$ and using Superset Relation is Compatible with Subset Product instead of Subset Relation is Compatible with Subset Product proves that $(3)$ implies $(1)$.