The inverse monoid associated to a group and the semidirect product of groups

In this paper, we construct an inverse monoid $Mleft( Gright) $ associatedto a given group $G$ by using the notion of the join of subgroups and then,by applying the left action of monoid $M$ on a semigroup $S$, we form asemigroup $Somega M$ on the set $Stimes M$. The finally result is to buildthe semi direct product of groups associated to the group action on ananother group.