Please help me about this question: I want to show that symplectic group $Sp(2n,q)$ has an element of order $q^n-1$.
2026-02-23 17:24:58.1771867498
find a special element of $Sp(2n,q)$
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Hint: Let $(V,f)$ be $2n$ dimensional symplectic vector space on $|\mathbb F|=q$. There is a basis such that the matrix of $f$ is $\left[ \begin{array}{cr} 0 & I \\ -I & 0\end{array} \right]$. It is not hard to see that for every $A\in GL(n,\mathbb F)$, the matrix $\left[ \begin{array}{cr} A & 0 \\ 0 & (A^{-1})^t\end{array} \right]$ belongs to $Sp(2n,q)$.