Exercise 4, § 4

Let $A$ and $B$ be two subgroups of a group $G$.

(a) Show that the least subgroup containing $A$ and $B$ (that is the subgroup generated by $A \cup B$) is identical with the set of compositions of the sequences $(x_i)_{1 \leq i \leq 2n+1}$ of an (arbitrary) odd number of elements such that $x_i \in A$ for $i$ odd and $x_i \in B$ for $i$ even.

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

(c) If $A$ and $B$ are permutable and $C$ is a subgroup containing $A$, $A$ is permutable with $B \cap C$ and $A(B \cap C) = C \cap (AB)$.