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

if and only if:

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

Proof

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:

\(\ds y \circ x\)

\(=\)

\(\ds e \circ \paren {y \circ x}\)

Group Axiom $\text G 2$: Existence of Identity Element

\(\ds \)

\(=\)

\(\ds \paren {x^{-1} \circ x} \circ \paren {y \circ x}\)

Group Axiom $\text G 3$: Existence of Inverse Element

\(\ds \)

\(=\)

\(\ds x^{-1} \circ \paren {x \circ y} \circ x\)

Group Axiom $\text G 1$: Associativity

\(\ds \leadsto \ \ \)

\(\ds y \circ x\)

\(\in\)

\(\ds x^{-1} \circ S \circ x\)

$x \circ y \in S$

\(\ds \leadsto \ \ \)

\(\ds y \circ x\)

\(\in\)

\(\ds S\)

by hypothesis: Definition 3 of Normal Subset

$\Box$

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:

\(\ds \forall x \in S: \, \)

\(\ds e \circ x \circ e\)

\(\in\)

\(\ds S\)

Group Axiom $\text G 2$: Existence of Identity Element

\(\ds \leadsto \ \ \)

\(\ds \forall x \in S: \, \)

\(\ds \paren {g \circ g^{-1} } \circ x \circ \paren {g \circ g^{-1} }\)

\(\in\)

\(\ds S\)

Group Axiom $\text G 3$: Existence of Inverse Element

\(\ds \leadsto \ \ \)

\(\ds \forall x \in S: \, \)

\(\ds g \circ \paren {g^{-1} \circ x \circ g \circ g^{-1} }\)

\(\in\)

\(\ds S\)

Group Axiom $\text G 1$: Associativity

\(\ds \leadsto \ \ \)

\(\ds \forall x \in S: \, \)

\(\ds \paren {g^{-1} \circ x \circ g \circ g^{-1} } \circ g\)

\(\in\)

\(\ds S\)

by hypothesis: Definition 5 of Normal Subset

\(\ds \leadsto \ \ \)

\(\ds \forall x \in S: \, \)

\(\ds \paren {g^{-1} \circ x \circ g} \circ \paren {g^{-1} \circ g}\)

\(\in\)

\(\ds S\)

Group Axiom $\text G 1$: Associativity

\(\ds \leadsto \ \ \)

\(\ds \forall x \in S: \, \)

\(\ds \paren {g^{-1} \circ x \circ g} \circ e\)

\(\in\)

\(\ds S\)

Definition of Inverse Element

\(\ds \leadsto \ \ \)

\(\ds \forall x \in S: \, \)

\(\ds g^{-1} \circ x \circ g\)

\(\in\)

\(\ds S\)

Definition of Identity Element

\(\ds \leadsto \ \ \)

\(\ds g^{-1} \circ S \circ g\)

\(\subseteq\)

\(\ds S\)

Definition of Subset Product

$\blacksquare$