## 2016年5月28日土曜日

### 数学 – 群 - 剰余類分解(左剰余類と右剰余類)

• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

$\begin{array}{l}{a}_{1},{a}_{2}\in G\\ {a}_{1}H={a}_{2}H\\ {a}_{1}{}^{-1}{a}_{2},{a}_{2}{}^{-1}{a}_{1}\in H\\ x\in H{a}_{1}^{-1}\\ h\in H\\ x=h{a}_{1}{}^{-1}\\ x{a}_{2}=h{a}_{1}{}^{-1}{a}_{2}\\ h\text{'}\in H\\ x{a}_{2}=h\text{'}\\ x{a}_{2}{a}_{2}{}^{-1}=h\text{'}{a}_{2}{}^{-1}\\ x=h\text{'}{a}_{2}{}^{-1}\\ x\in H{a}_{2}{}^{-1}\\ H{a}_{2}{}^{-1}\subset H{a}_{1}{}^{-1}\\ x\in H{a}_{2}{}^{-1}\\ x=h{a}_{2}{}^{-1}\\ x{a}_{1}=h{a}_{2}{}^{-1}{a}_{1}\\ h\text{'}\in H\\ x{a}_{1}=h\text{'}\\ x{a}_{1}{a}_{1}{}^{-1}=h\text{'}{a}_{1}{}^{-1}\\ x=h\text{'}{a}_{1}{}^{-1}\\ x\in H{a}_{1}{}^{-1}\\ H{a}_{2}{}^{-1}\subset H{a}_{1}{}^{-1}\\ H{a}_{1}{}^{-1}=H{a}_{2}{}^{-1}\\ \\ y\in {Q}_{r}\\ a\in G\\ y=H{a}^{-1}\\ aH\in {G}_{l}\\ {y}_{1},{y}_{2}\in {Q}_{r}\\ {a}_{1},{a}_{2}\in G\\ {y}_{1}=H{a}_{1}{}^{-1},{y}_{2}=H{a}_{2}{}^{-1}\\ H{a}_{1}{}^{-1}=H{a}_{2}{}^{-1}\\ {a}_{1}{}^{-1}{a}_{2},{a}_{2}{}^{-1}{a}_{1}\in H\\ x\in {a}_{1}H\\ h\in H\\ x={a}_{1}h\\ {a}_{2}{}^{-1}x={a}_{2}{}^{-1}{a}_{1}h\\ h\text{'}\in H\\ {a}_{2}{}^{-1}x=h\text{'}h\\ h\text{'}\text{'}\in H\\ {a}_{2}{}^{-1}x=h\text{'}\text{'}\\ {a}_{2}{a}_{2}{}^{-1}x={a}_{2}h\text{'}\text{'}\\ x={a}_{2}h\text{'}\text{'}\\ x\in {a}_{2}H\\ {a}_{1}H\subset {a}_{2}H\\ x\in {a}_{2}H\\ h\in H\\ x={a}_{2}h\\ {a}_{1}{}^{-1}x={a}_{1}{}^{-1}{a}_{2}h\\ h\text{'}\in H\\ {a}_{1}{}^{-1}x=h\text{'}h\\ h\text{'}\text{'}\in H\\ {a}_{1}{}^{-1}x=h\text{'}\text{'}\\ x={a}_{1}h\text{'}\text{'}\\ x\in {a}_{1}H\\ {a}_{2}H\subset {a}_{1}H\\ {a}_{1}H={a}_{2}H\end{array}$