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 defining ingredient::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 $$q$$.

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 $$PSp(2,2), PSp(2,3), PSp(4,2)$$.