Symplectic group is quasisimple

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) uses::Symplectic group is perfect except in the three cases listed above.
 * 2) uses::Projective symplectic group is simple except in the three cases listed above.

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