数学 – 群 - 剰余類分解(左剰余類と右剰余類)

学習環境

代数系入門 (松坂和夫(著)、岩波書店)の第2章(群)、4(剰余類分解)、問1.を取り組んでみる。

問1.

\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' \in H \\ x a_{2} = h' \\ x a_{2} a_{2}^{- 1} = h' a_{2}^{- 1} \\ x = h' 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' \in H \\ x a_{1} = h' \\ x a_{1} a_{1}^{- 1} = h' a_{1}^{- 1} \\ x = h' 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} \\ a H \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' \in H \\ a_{2}^{- 1} x = h' h \\ h'' \in H \\ a_{2}^{- 1} x = h'' \\ a_{2} a_{2}^{- 1} x = a_{2} h'' \\ x = a_{2} h'' \\ 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' \in H \\ a_{1}^{- 1} x = h' h \\ h'' \in H \\ a_{1}^{- 1} x = h'' \\ x = a_{1} h'' \\ x \in a_{1} H \\ a_{2} H \subset a_{1} H \\ a_{1} H = a_{2} H \end{array}

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