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:

“In the order of literature, as in others, there is no act that is not the coronation of an infinite series of causes and the source of an infinite series of effects.”
—Jorge Luis Borges (1899–1986)

“Woman ... cannot be content with health and agility: she must make exorbitant efforts to appear something that never could exist without a diligent perversion of nature. Is it too much to ask that women be spared the daily struggle for superhuman beauty in order to offer it to the caresses of a subhumanly ugly mate?”
—Germaine Greer (b. 1939)

“As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”
—Margaret Atwood (b. 1939)

“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)