Jordan's Totient Function - Order of Matrix Groups

Order of Matrix Groups

The general linear group of matrices of order m over Z_n has order

$|\operatorname{GL}(m,\mathbf{Z}_n)|=n^{\frac{m(m-1)}{2}}\prod_{k=1}^m J_k(n).$

The special linear group of matrices of order m over Z_n has order

$|\operatorname{SL}(m,\mathbf{Z}_n)|=n^{\frac{m(m-1)}{2}}\prod_{k=2}^m J_k(n).$

The symplectic group of matrices of order m over Z_n has order

$|\operatorname{Sp}(2m,\mathbf{Z}_n)|=n^{m^2}\prod_{k=1}^m J_{2k}(n).$

The first two formulas were discovered by Jordan.

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