数学 – 群 - 群とその例(公理、単位元、逆元)

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代数系入門 (松坂和夫(著)、岩波書店)の第2章(群)、2(群とその例)、問3.を解いてみる。

問3.

\begin{array}{l} a \in G \\ a_{0} \in G \\ e_{0} \in G \\ a_{0} e_{0} = a_{0} \\ v_{0} \in G \\ v_{0} a_{0} = a \\ a e_{0} = (v_{0} a_{0}) e_{0} \\ = v_{0} (a_{0} e_{0}) \\ = v a_{0} \\ = a \\ a e_{0} = a \\ a_{1} \in G \\ e_{1} \in G \\ e_{1} a_{1} = a_{1} \\ u_{0} \in G \\ a_{1} u_{0} = a \\ e_{1} a_{1} \\ e_{1} a = e_{1} (a_{1} u_{0}) \\ = (e_{1} a_{1}) u_{0} \\ = a_{1} u_{0} \\ = a \\ e_{1} a = a \\ e_{0} = e_{1} e_{0} = e_{1} \\ e_{0} = e_{1} \\ e_{0} = e_{1} = e \\ e a = a e = a \\ w \in G \\ a w = e \\ x \in G \\ x a = e \\ w = e w \\ = (x a) w \\ = x (a w) \\ = x \\ w = x \\ a^{- 1} = w = x \end{array}

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