# Projective symplectic group

From Groupprops

## Contents

## Definition

Let be a positive integer and be any field. The **projective symplectic group** or is defined in the following equivalent ways:

- It is the quotient of the symplectic group by the scalar matrices in the group.
- It is the inner automorphism group of the symplectic group , i.e., the quotient of that group by its center.

For a prime power , we denote by the group where is the (unique up to isomorphism) field of size .

### Chevalley notation

The projective symplectic group is the Chevalley group of type C, denoted . Note that the degree parameter when describing it as a Chevalley group is *half* the size of the matrices.

For a prime power , we denote by the group where is the (unique up to isomorphism) field of size .

## Facts

### Collisions with other Chevalley groups

- is isomorphic to as well as to , which is the projective special linear group of degree two .
- is isomorphic to , where denotes the Chevalley group of type B, and arises as a subgroup of the orthogonal group.

### Simplicity

- Projective symplectic group is simple except the cases .