Abstract Algebra/Group Theory/Group/Ga = G

= Theorem = Let G be any Group.

Let $$ Ga = \lbrace g \ast a \; | \;  g \in G \rbrace $$


 * $$ \forall \; a \in G: Ga = G $$

= Proof = Part A. $$ \color{RawSienna}Ga \subseteq G$$


 * {|Style = "width:60%"

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 * 0. Choose $$ {\color{OliveGreen}a} \in G$$
 * 1. Choose $$x \in G{\color{OliveGreen}a} = \lbrace g \ast {\color{OliveGreen}a} \; | \; g \in G \rbrace$$
 * 1. Choose $$x \in G{\color{OliveGreen}a} = \lbrace g \ast {\color{OliveGreen}a} \; | \; g \in G \rbrace$$

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 * 2. $$ \exists \; g \in G: x = g \ast {\color{OliveGreen}a} $$
 * 2. $$ \exists \; g \in G: x = g \ast {\color{OliveGreen}a} $$

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 * 1.
 * 3. $$ g \ast {\color{OliveGreen}a} \in G $$
 * 3. $$ g \ast {\color{OliveGreen}a} \in G $$

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 * closure of G, $$ g, a \in G $$
 * 4. $$ x = g \ast {\color{OliveGreen}a} \in G $$
 * 4. $$ x = g \ast {\color{OliveGreen}a} \in G $$


 * 2,
 * }

Part B. $$ \color{RawSienna}G \subseteq Ga$$


 * {|Style = "width:60%"

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 * 5. Choose $$ {\color{OliveGreen}a} \in G $$
 * 6. $$ \exists \; {\color{BrickRed}a^{-1}} \in G: {\color{BrickRed}a^{-1}} \ast {\color{OliveGreen}a} = e_{G} $$
 * 6. $$ \exists \; {\color{BrickRed}a^{-1}} \in G: {\color{BrickRed}a^{-1}} \ast {\color{OliveGreen}a} = e_{G} $$

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 * definition of inverse
 * 7. Choose $$ y \in G $$
 * 7. Choose $$ y \in G $$

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 * 8. $$ y \ast {\color{BrickRed}a^{-1}} \in G $$
 * 8. $$ y \ast {\color{BrickRed}a^{-1}} \in G $$

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 * closure of G, and, y, a−1 are in G
 * 9. $$ (y \ast {\color{BrickRed}a^{-1}}) \ast {\color{OliveGreen}a} \in G{\color{OliveGreen}a} $$
 * 9. $$ (y \ast {\color{BrickRed}a^{-1}}) \ast {\color{OliveGreen}a} \in G{\color{OliveGreen}a} $$

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 * definition of G a
 * 10. $$ y \ast ({\color{BrickRed}a^{-1}} \ast {\color{OliveGreen}a}) \in G{\color{OliveGreen}a} $$
 * 10. $$ y \ast ({\color{BrickRed}a^{-1}} \ast {\color{OliveGreen}a}) \in G{\color{OliveGreen}a} $$

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 * associativity on G (not Ga)
 * 11. $$ y \ast e_{G} \in G{\color{OliveGreen}a} $$
 * 11. $$ y \ast e_{G} \in G{\color{OliveGreen}a} $$

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 * eG is identity of G
 * 12. $$ y \in G{\color{OliveGreen}a} $$
 * 12. $$ y \in G{\color{OliveGreen}a} $$


 * }
 * }

Part C. $$ \color{RawSienna}Ga = G$$