Isomorphism between symplectic and projective symplectic group in characteristic two

Statement
Suppose $$K$$ is a field and $$m$$ is a positive integer. The following are equivalent:


 * 1) $$K$$ has characteristic two.
 * 2) The symplectic group $$Sp(2m,K)$$ is a centerless group.
 * 3) The natural quotient map from the symplectic group $$Sp(2m,K)$$ to the projective symplectic group $$PSp(2m,K)$$ is an isomorphism of groups.
 * 4) The  symplectic group $$Sp(2m,K)$$ and projective symplectic group $$PSp(2m,K)$$ are isomorphic groups.

In any other characteristic, the center of $$Sp(2m,K)$$ is cyclic of order two.