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how to prove the simplicity of unitary group over finite fields

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    $\begingroup$ The question you ask in the body is not the same as the question you ask in the title. Please fix this. $\endgroup$
    – Pete L. Clark
    Commented Nov 28, 2010 at 7:14

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A proof that special unitary groups over a field $F$ are generated by transvections -- except when $n = 3$ and $\# F = 4$ in which case the result is not true -- is given in Larry Grove's text: see Theorem 11.15 on page 104.

The simplicity of the projective special unitary group $PSU(V)$ -- except when $(\operatorname{dim} V, \# F)$ is one of $(2,4)$, $(2,9)$ and $(3,4)$ in which cases the result is not true -- is Theorem 11.26 on page 108 of Grove's text.

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