Let $G$ be a group, $S$ its finite set of generators. Denote $G^+$ the subsemigroup of $G$ generated by $S$. Let $g\in G$, $H<G$ the subgroup generated by $g$ and let $g\_1,...,g\_k\in G$ be a finite collection. Suppose $G^+\subset g\_1H\cup\cdots\cup g\_kH$. Is it true that $H$ is of finite index in $G$?


**Edit:** as pointed out by Tom Goodwillie, the question is true without any assumption on $H$.



Group with virtually cyclic positive part.
Restating the question as I understand it:


Let $G$ be a group, $S\subset G$ a finite set of generators, $H$ a cyclic subgroup of $G$. Suppose that there is a finite collection of elements $g\_j\in G$ such that every product of finitely many elements of $S$ belongs to one of the cosets $g\_jH$. Is it true that $H$ has finite index in $G$?


The answer is yes, even without assuming $H$ cyclic. The group $G$ acts on the set $G/H$ of all cosets of $H$ in the usual fashion. The smallest subset of $G/H$ that contains the element $H$ and is closed under the action of each element of $S$ is finite. Therefore each element of $S$ maps this set to itself bijectively. Therefore so does the inverse of each element of $S$. Therefore so does each element of $G$. But the action of $G$ on $G/H$ is transitive, so it follows that this finite set is the whole of $G/H$, and $H$ has finite index.


