Equivalence of Definitions of Normal Subset/3 and 4 imply 2
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Theorem
Let $\struct {G, \circ}$ be a group.
Let $S \subseteq G$.
Let $S$ be a normal subset of $G$ by Definition 3 and Definition 4.
Then $S$ is a normal subset of $G$ by Definition 2.
Proof
By Equivalence of Definitions of Normal Subset: 3 iff 4, $S$ being a normal subset of $G$ by Definition 3 and Definition 4 implies that the following hold:
- $(1)\quad \forall g \in G: g \circ S \circ g^{-1} \subseteq S$
- $(2)\quad \forall g \in G: g^{-1} \circ S \circ g \subseteq S$
- $(3)\quad \forall g \in G: S \subseteq g \circ S \circ g^{-1}$
- $(4)\quad \forall g \in G: S \subseteq g^{-1} \circ S \circ g$
By $(1)$ and $(3)$ and definition of set equality:
- $\forall g \in G: g \circ S \circ g^{-1} = S$
By $(2)$ and $(4)$ and definition of set equality:
- $\forall g \in G: g^{-1} \circ S \circ g = S$
$\blacksquare$