Symplectic group is quasisimple

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

Statement

Suppose $m$ is a positive integer and $k$ is a field. Then, the symplectic group $Sp(2m,k)$ is a quasisimple group except in the cases $m=1,|k|=2$, $m=1,|k|=3$ and $m=2,|k|=2$. In other words, the only exceptions are $Sp(2,2),Sp(2,3),Sp(4,2)$.

A quasisimple group is a perfect group whose inner automorphism group is a simple group.

Related facts

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

Facts used

1. Symplectic group is perfect except in the three cases listed above.
2. Projective symplectic group is simple except in the three cases listed above.

Proof

The proof follows directly from facts (1) and (2).