2 problems

Let $G$ be a group. Assume $H_1,\ldots,H_k$ are proper subgroups of $G$, with $k \geq 2$, and let $x_1,\ldots,x_k$ be elements of $G$ such that … and…

Let $\Sigma$ be a finite alphabet, and let $\Sigma^*$ be the free monoid generated by $\Sigma$. For each $1\leq i\leq s$, let $M_i$ be a finite, bi-deterministic and complete autom…