# Symplectic group is quasisimple

From Groupprops

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 (?)).

## Contents

## Statement

Suppose is a positive integer and is a field. Then, the symplectic group is a quasisimple group except in the cases , and . In other words, the only exceptions are .

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

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

## Proof

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