Omissible extensions of $SL_2(k)$ where $k$ is a field of positive characteristic

Abstract

A normal subgroup $N$ of a group $G$ is said to be an‎ <em>omissible</em> subgroup of $G$ if it has the following property‎: ‎whenever $Xleq G$ is such that $G=XN$‎, ‎then $G=X$‎. ‎In this note we construct various groups $G$‎, ‎each of which has an omissible subgroup $Nneq 1$ such that $G/Ncong SL_2(k)$ where $k$ is a field of positive characteristic‎.