习题 4, § 4

令$A$和$B$为群$G$中的两个子群。

(a) Show that the least 子群包含 $A$ and $B$ (that is 子群 generated by $A \cup B$) is identical with 集合 of 合成 of 序列 $(x_i)_{1 \leq i \leq 2n+1}$ of an (arbitrary) odd 数 of 元素 such that $x_i \in A$ for $i$ odd and $x_i \in B$ for $i$ even.

(b) For $AB$ to be 子群 of $G$ (in which case it is 子群 generated by $A \cup B$), it is necessary and sufficient that $A$ and $B$ be permutable, that is that $AB = BA$.

(c) 如果$A$和$B$是可置换的，并且$C$是子群包含$A$，则$A$可以用$B \cap C$和$A(B \cap C) = C \cap (AB)$进行置换。