Symplectic group is quasisimple

From Groupprops
Jump to: navigation, search
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

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).