Perfectness of finite symplectic groups
= Perfectness of finite symplectic groups
For $q>3$, embedded perfect $SL_2(q)$ groups contain every <symplectic transvection>. For $q=3,m\ge2$, four coefficient-one transvections on the lines of an isotropic plane multiply to the identity; their abelianized class is annihilated by 3 and 4. For $q=2,m\ge3$, seven transvections on an isotropic three-space multiply to the identity; their class is annihilated by 2 and 7. Thus all nonexceptional cases are perfect.