Isomorphism between symplectic and projective symplectic group in characteristic two

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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.