Order formulas for linear groups

This article gives a list of formulas for the orders of the general linear group of finite degree $n\geq 1$ and some other related groups, both for a finite field of size $q$ and for related rings.

For a finite field of size $q$

Explanation for order of general linear group

We describe here the reasoning behind the formula for the order of the general linear group $GL(n,q)=GL(n,\mathbb {F} _{q})$.

The order equals the number of invertible $n\times n$ matrices with entries over $\mathbb {F} _{q}$. The set of uch matrices is in correpsondence with the set of ordered bases for $\mathbb {F} _{q}^{n}$, the $n$-dimensional vector space over $\mathbb {F} _{q}$ (for instance, we can identify the columns of the matrix with the vectors in the ordered basis). Thus, it suffices to count the number of possible ordered bases for $\mathbb {F} _{q}^{n}$.

For the first vector of the ordered basis, there are $q^{n}-1$ possible choices (all nonzero vectors work). For the second vector, there are $q^{n}-q$ choices (all vectors that are not in the span of the first vector work). After the first $i$ basis vectors are chosen, the number of possibilities for the next basis vector are $q^{n}-q^{i}$, because $q^{i}$ is the size of the subspace spanned by the first $i$ basis vectors. By the product rule in combinatorics, we get that the total number of possibilities is:

$\prod _{i=1}^{n}(q^{n}-q^{i})=(q^{n}-1)(q^{n}-q)\dots (q^{n}-q^{n-1})$

Formulas

In the formulas below, the field size is $q$ and the degree (order of matrices involved, dimension of vector space being acted upon) is $n$. The characteristic of the field is a prime number $p$. $q$ is a prime power with underlying prime $p$. We let $r=\log _{p}q$, so $q=p^{r}$ and $r$ is a nonnegative integer.

Group	Symbolic notation	Order formula	Order formula (powers of $q$ taken out)	Order formula (maximally factorized)	Degree as polynomial in $q$ (same as algebraic dimension)	Multiplicity of factor $q$	Multiplicity of factor $q-1$	Quick explanation for order
general linear group	$GL(n,q)$ or $GL(n,\mathbb {F} _{q})$	$\prod _{i=0}^{n-1}(q^{n}-q^{i})$	$q^{\binom {n}{2}}\prod _{i=0}^{n-1}(q^{n-i}-1)$	$q^{\binom {n}{2}}\prod _{d=1}^{n}(\Phi _{d}(q))^{\lfloor n/d\rfloor }$	$n^{2}$	${\binom {n}{2}}={\frac {n(n-1)}{2}}$	$n$	See full explanation above.
special linear group	$SL(n,q)$ or $SL(n,\mathbb {F} _{q})$	${\frac {\prod _{i=0}^{n-1}(q^{n}-q^{i})}{q-1}}$	$q^{\binom {n}{2}}\prod _{i=0}^{n-2}(q^{n-i}-1)$	$q^{\binom {n}{2}}(q-1)^{n-1}\prod _{d=2}^{n}(\Phi _{d}(q))^{\lfloor n/d\rfloor }$	$n^{2}-1$	${\binom {n}{2}}={\frac {n(n-1)}{2}}$	$n-1$	[SHOW MORE] The group is the kernel of the determinant map, a surjective homomorphism from $GL(n,q)$ to $\mathbb {F} _{q}^{\ast }$. Its order is thus $|GL(n,q)|/|\mathbb {F} _{q}^{\ast }|$.
projective general linear group	$PGL(n,q)$ or $PGL(n,\mathbb {F} _{q})$	${\frac {\prod _{i=0}^{n-1}(q^{n}-q^{i})}{q-1}}$	$q^{\binom {n}{2}}\prod _{i=0}^{n-2}(q^{n-i}-1)$	$q^{\binom {n}{2}}(q-1)^{n-1}\prod _{d=2}^{n}(\Phi _{d}(q))^{\lfloor n/d\rfloor }$	$n^{2}-1$	${\binom {n}{2}}={\frac {n(n-1)}{2}}$	$n-1$	[SHOW MORE] The group is the quotient of $GL(n,q)$ by the center, which is the subgroup of scalar matrices, and is isomorphic to $\mathbb {F} _{q}^{\ast }$. Its order is thus $|GL(n,q)|/|\mathbb {F} _{q}^{\ast }|$.
projective general linear group	$PGL(n,q)$ or $PGL(n,\mathbb {F} _{q})$	${\frac {\prod _{i=0}^{n-1}(q^{n}-q^{i})}{(q-1)\operatorname {gcd} (n,q-1)}}$	$q^{\binom {n}{2}}{\frac {\prod _{i=0}^{n-2}(q^{n-i}-1)}{\operatorname {gcd} (n,q-1)}}$	$q^{\binom {n}{2}}{\frac {(q-1)^{n-1}\prod _{d=2}^{n}(\Phi _{d}(q))^{\lfloor n/d\rfloor }}{\operatorname {gcd} (n,q-1)}}$	$n^{2}-1$ (ignoring gcd term)	${\binom {n}{2}}={\frac {n(n-1)}{2}}$	$n-1$ (ignoring gcd term)	[SHOW MORE] The group is the quotient of $SL(n,q)$ by its intersection with the center, which is the subgroup of scalar matrices of determinant 1, and this group is cyclic of order $\operatorname {gcd} (n,q-1)$. Its order is thus $|SL(n,q)|/\operatorname {gcd} (n,q-1)$.