Q: Subgroup generated by a finite family of subgroups Let $(H_{i})_{i\in I}$ be a finite family of subgroups of a group $G$. Then
$$\bigvee_{i\in I} H_{i}=\bigcup_{n\in\mathbb{N}_{\geq1}}\{y\ |\ (\exists x)(x=(x_i)\in(\cup H_i)^{[1,n]}\ \land y=x_1\dots x_n)\}$$
is the subgroup generated by the family ($H_{i}$). 
Under what circumstances, for a $y\in\vee H_i$, can one match the indices of the $(H_i)$ and the $(x_i)$ in a one-to-one manner?
