# Projective symplectic group

## Definition

Let $m$ be a positive integer and $K$ be any field. The projective symplectic group $PSp(2m,k)$ or $PSp_{2m}(K)$ is defined in the following equivalent ways:

• It is the quotient of the symplectic group $Sp(2m,K)$ by the scalar matrices in the group.
• It is the inner automorphism group of the symplectic group $Sp(2m,K)$, i.e., the quotient of that group by its center.

For a prime power $q$, we denote by $PSp(2m,q)$ the group $PSp(2m,\mathbb{F}_q)$ where $\mathbb{F}_q$ is the (unique up to isomorphism) field of size $q$.

### Chevalley notation

The projective symplectic group $PSp(2m,K)$ is the Chevalley group of type C, denoted $C_m(K)$. Note that the degree parameter when describing it as a Chevalley group is half the size of the matrices.

For a prime power $q$, we denote by $C_m(q)$ the group $C_m(\mathbb{F}_q)$ where $\mathbb{F}_q$ is the (unique up to isomorphism) field of size $q$.

## Facts

### Collisions with other Chevalley groups

• $C_1(K)$ is isomorphic to $B_1(K)$ as well as to $A_1(K)$, which is the projective special linear group of degree two $PSL(2,K)$.
• $C_m(2^n)$ is isomorphic to $B_m(2^n)$, where $B_m$ 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 $PSp(2,2), PSp(2,3), PSp(4,2)$.