orders and structure of classical groups

Theorem 1.

\begin{array}[]{cc}|SL(d,q)|=q^{\binom{d}{2}}\prod_{i=2}^{d}(q^{i}-1),&|PSL(d,% q)|=\frac{|SL(d,q)|}{(d,q-1)},\\ |GL(d,q)|=q^{\binom{d}{2}}\prod_{i=1}^{d}(q^{i}-1),&|PGL(d,q)|=\frac{|GL(d,q)|% }{q-1},\\ |\Gamma L(d,q)|=(q-1)q^{\binom{d}{2}}\prod_{i=1}^{d}(q^{i}-1),&|P\Gamma L(d,q)% |=\frac{|\Gamma L(d,q)|}{q-1}.\end{array}

A proof of this theorem can proceed in many directions which each explain more of the structure of the these groups so each approach is considered equally valid and presented here. We list them from most elementary to most theoretical.

1.

Elementary linear algebra proof (http://planetmath.org/ElementaryProofOfOrders)
2.

Proof from subgroups of $GL(V)$ (http://planetmath.org/TheoryFromOrdersOfClassicalGroups2)
3.

Proof from Lie theory/Chevally groups. [PENDING.]

Title	orders and structure of classical groups
Canonical name	OrdersAndStructureOfClassicalGroups
Date of creation	2013-03-22 15:56:45
Last modified on	2013-03-22 15:56:45
Owner	Algeboy (12884)
Last modified by	Algeboy (12884)
Numerical id	10
Author	Algeboy (12884)
Entry type	Topic
Classification	msc 11E57
Classification	msc 05E15