Projective symplectic group is simple

This article gives the statement, and possibly proof, of a particular group or type of group (namely, Projective symplectic group (?)) satisfying a particular group property (namely, Simple group (?)).

Statement

Suppose $m$ is a positive integer and $k$ is a field. Then, the projective symplectic group $PSp(2m,k)$ is a simple group except in the following three cases: $m=1$ and $k$ has two elements, $m=1$ and $k$ has three elements, $m=2$ and $k$ has two elements. In other words, the three exceptions are $PSp(2,2),PSp(2,3),PSp(4,2)$.

Related facts

• Symplectic group is perfect
• Symplectic group is quasisimple
• Projective special linear group is simple
• Special linear group is perfect
• Special linear group is quasisimple