数学 - 群論 - 部分群と生成系(部分集合, 可換)

代数系入門
(岩波書店)
松坂 和夫 (著)

学習環境

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代数系入門 (松坂 和夫(著)、岩波書店)の第2章(群論)、3(部分群と生成系)、問6.を解いてみる。

問6.

$\begin{array}{l}h\in H,h\text{'}\in H\text{'}\\ h=s_1···s_n\left(s_i\in S\vee s_i\in S^{-1}\right)\\ h\text{'}=s{\text{'}}_1···s{\text{'}}_m\left(s{\text{'}}_i\in S\text{'}\vee s_i\in S{\text{'}}^{-1}\right)\\ hh\text{'}=s_1···s_{n}s{\text{'}}_1···s{\text{'}}_m\\ \\ s\in S,s\text{'}\in S\text{'}\\ ss\text{'}=s\text{'}s\\ \\ s\in S,a\in S{\text{'}}^{-1}\\ sa=esa\\ =a{a}^{-1}sa\left(a^{-1}\in S\text{'}\right)\\ =as{a}^{-1}a\\ =as\\ \\ a\in S^{-1},s\text{'}\in S\text{'}\\ as\text{'}=s\text{'}s{\text{'}}^{-1}as\text{'}\left(s{\text{'}}^{-1}\in S\right)\\ =s\text{'}as{\text{'}}^{-1}s\text{'}\\ =s\text{'}ae\\ =s\text{'}a\\ \\ a\in S^{-1},b\in S{\text{'}}^{-1}\\ ab=eab\\ =b{b}^{-1}ab\left(a{b}^{-1}\in S\text{'}\right)\\ =ba{b}^{-1}b\\ =bae\\ =ba\\ \\ hh\text{'}=s{\text{'}}_1···s{\text{'}}_m{s}_1···s_n\\ =h\text{'}h\end{array}$