抽象代数/群论/群/Ga = G

定理

设 G 为任意群。

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

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

证明

部分 A. $\color {RawSienna}Ga\subseteq G$

0. 选择 ${\color {OliveGreen}a}\in G$
1. 选择 $x\in G{\color {OliveGreen}a}=\lbrace g\ast {\color {OliveGreen}a}\;|\;g\in G\rbrace$
2. $\exists \;g\in G:x=g\ast {\color {OliveGreen}a}$	1.
3. $g\ast {\color {OliveGreen}a}\in G$	G 的封闭性, $g,a\in G$
4. $x=g\ast {\color {OliveGreen}a}\in G$	2,

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

5. 选择 ${\color {OliveGreen}a}\in G$
6. $\exists \;{\color {BrickRed}a^{-1}}\in G:{\color {BrickRed}a^{-1}}\ast {\color {OliveGreen}a}=e_{G}$	逆元的定义
7. 选择 $y\in G$
8. $y\ast {\color {BrickRed}a^{-1}}\in G$	根据 G 的封闭性，y 和 a−1 都属于 G
9. $(y\ast {\color {BrickRed}a^{-1}})\ast {\color {OliveGreen}a}\in G{\color {OliveGreen}a}$	根据 Ga 的定义
10. $y\ast ({\color {BrickRed}a^{-1}}\ast {\color {OliveGreen}a})\in G{\color {OliveGreen}a}$	根据 G（而非 Ga）的结合律
11. $y\ast e_{G}\in G{\color {OliveGreen}a}$	eG 是 G 的单位元
12. $y\in G{\color {OliveGreen}a}$

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C 部分. $\color {RawSienna}Ga=G$

$Ga=G\,$

$G\subseteq Ga$ 且 $Ga\subseteq G$,