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.

Read more about this topic: Jordan's Totient Function

Famous quotes containing the words order of, order, matrix and/or groups:

“My ideas are a curse.
They spring from a radical discontent
with the awful order of things.
I play clown. I play carpenter. I play nurse.
I play witch.”
—Anne Sexton (1928–1974)

“Ever since surgery began, man’s destiny has been to suffer, in order that he might be cured. And no one can change that, gentlemen.”
—Jean Scott Rogers. Robert Day. Mr. Blount (Frank Pettingell)

“In all cultures, the family imprints its members with selfhood. Human experience of identity has two elements; a sense of belonging and a sense of being separate. The laboratory in which these ingredients are mixed and dispensed is the family, the matrix of identity.”
—Salvador Minuchin (20th century)

“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)