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$emph{omissible}$ 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.

Keywords

• Omissible subgroup
• special linear group
• Frattini extension
• locally (soluble-by-finite) group