Order formulas for linear groups of degree two

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

For the related formulas for degrees other than two, as well as for more detailed explanations of the order formulas, see order formulas for linear groups.

For a finite field of size $q$

Formulas

In the formulas below, the field size is $q$. 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 (maximally factorized)	Order formula (expanded)	Degree as polynomial in $q$ (same as algebraic dimension)	Quick explanation for order
general linear group of degree two	$GL(2,q)$ or $GL(2,\mathbb {F} _{q})$	$(q^{2}-1)(q^{2}-q)$	$q(q-1)^{2}(q+1)$	$q^{4}-q^{3}-q^{2}+q$	4	[SHOW MORE] The size is the number of ordered pairs of linearly independent vectors in a two-dimensional vector space over $\mathbb {F} _{q}$. For the first vector, there are $q^{2}-1$ possible choices (all nonzero vectors work). For the second vector, there are $q^{2}-q$ choices (all vectors that are not in the span of the first vector work). The product rule of combinatorics gives a total of $(q^{2}-1)(q^{2}-q)$ possibilities.
projective general linear group of degree two	$PGL(2,q)$ or $PGL(2,\mathbb {F} _{q})$	$q(q^{2}-1)$	$q(q-1)(q+1)$	$q^{3}-q$	3	[SHOW MORE] This group is the quotient of $GL(2,q)$ by the subgroup of scalar matrices (see also center of general linear group is group of scalar matrices over center) which is isomorphic to $\mathbb {F} _{q}^{\ast }$ and has order $(q-1)$. The order of $PGL(2,q)$ is thus $|GL(2,q)|/|\mathbb {F} _{q}^{\ast }|$ which becomes $(q^{2}-1)(q^{2}-q)/(q-1)=q(q^{2}-1)$.
special linear group of degree two	$SL(2,q)$ or $SL(2,\mathbb {F} _{q})$	$q(q^{2}-1)$	$q(q-1)(q+1)$	$q^{3}-q$	3	[SHOW MORE] This is the kernel of the determinant homomorphism $GL(2,q)\to \mathbb {F} _{q}^{\ast }$. The determinant homomorphism is surjective, because every $a\in \mathbb {F} _{q}^{\ast }$ is the determinant of ${\begin{pmatrix}a&0\\0&1\\\end{pmatrix}}\in GL(2,q)$. Thus, the order of the kernel is $|GL(2,q)|/|\mathbb {F} _{q}^{\ast }|$ which becomes $(q^{2}-1)(q^{2}-q)/(q-1)=q(q^{2}-1)$.
projective special linear group of degree two	$PSL(2,q)$ or $PSL(2,\mathbb {F} _{q})$	$q(q^{2}-1)/\operatorname {gcd} (2,q-1)$ $q(q^{2}-1)/2$ for $q$ odd $q(q^{2}-1)$ for $q$ even	$q(q-1)(q+1)/\operatorname {gcd} (2,q-1)$ $q(q-1)(q+1)/2$ for $q$ odd $q(q-1)(q+1)$ for $q$ even	$(q^{3}-q)/\operatorname {gcd} (2,q-1)$ $(q^{3}-q)/2$ for $q$ odd $q^{3}-q$ for $q$ even	3	[SHOW MORE] This group is the quotient of $SL(2,q)$ by the subgroup of scalar matrices in $SL(2,q)$. For a matrix ${\begin{pmatrix}a&0\\0&a\\\end{pmatrix}}$ to be in $SL(2,q)$, we must have $a^{2}=1$ so $a=\pm 1$. If $q$ (and hence $p$) is odd, then there are two solutions so the kernel has order two, giving $|PSL(2,q)|=|SL(2,q)|/2$. If $q$ is even (so $p=2$), then the kernel is trivial and $|PSL(2,q)|=|SL(2,q)|$.
general semilinear group of degree two	$\Gamma L(2,q)$ or $\Gamma L(2,\mathbb {F} _{q})$	$r(q^{2}-1)(q^{2}-q)$	$rq(q-1)^{2}(q+1)$	$rq^{4}-rq^{3}-rq^{2}+rq$	4	[SHOW MORE] This group is a semidirect product of $GL(2,q)$ (order $(q^{2}-1)(q^{2}-q)$) by a cyclic Galois group of order $r$ (the Galois group of $\mathbb {F} _{q}$ over $\mathbb {F} _{p}$. We use order of semidirect product is product of orders.
projective semilinear group of degree two	$P\Gamma L(2,q)$ or $P\Gamma L(2,\mathbb {F} _{q})$	$rq(q^{2}-1)$	$rq(q-1)(q+1)$	$rq^{3}-rq$	3	[SHOW MORE] This group is a semidirect product of $PGL(2,q)$ by a cyclic Galois group of order $r$ (the Galois group of $\mathbb {F} _{q}$ over $\mathbb {F} _{p}$. We use order of semidirect product is product of orders.
general affine group of degree two	$GA(2,q)$ or $AGL(2,q)$ or $GA(2,\mathbb {F} _{q})$ or $AGL(2,\mathbb {F} _{q})$	$q^{2}(q^{2}-1)(q^{2}-q)$	$q^{3}(q-1)^{2}(q+1)$	$q^{6}-q^{5}-q^{4}+q^{3}$	6	[SHOW MORE] This group is a semidirect product of the vector space of dimension two over $\mathbb {F} _{q}$ (order $q^{2}$) by the group $GL(2,q)$ (order $(q^{2}-1)(q^{2}-q)$). We use order of semidirect product is product of orders.
special affine group of degree two	$SA(2,q)$ or $ASL(2,q)$ or $SA(2,\mathbb {F} _{q})$ or $ASL(2,\mathbb {F} _{q})$	$q^{2}(q^{3}-q)$	$q^{3}(q-1)(q+1)$	$q^{5}-q^{3}$	5	[SHOW MORE] This group is a semidirect product of the vector space of dimension two over $\mathbb {F} _{q}$ (order $q^{2}$) by the group $SL(2,q)$ (order $q^{3}-q$). We use order of semidirect product is product of orders.

Particular cases

The links are to the actual groups, which are not explicitly specified in order to save space.

Field size $q$	Field characteristic $p$	$r$ so that $q=p^{r}$	$|GL(2,q)|$ $=(q^{2}-1)(q^{2}-q)$	$|PGL(2,q)|$ $=q^{3}-q$	$|SL(2,q)|$ $=q^{3}-q$	$|PSL(2,q)|$ $=(q^{3}-q)/\operatorname {gcd} (2,q-1)$	$|\Gamma L(2,q)|=$ $r(q^{2}-1)(q^{2}-q)$	$|P\Gamma L(2,q)|=$ $r(q^{3}-q)$	$|GA(2,q)|=$ $q^{2}(q^{2}-1)(q^{2}-q)$
2	2	1	6	6	6	6	6	6	24
3	3	1	48	24	24	12	48	24	432
4	2	2	180	60	60	60	360	120	2880
5	5	1	480	120	120	60	480	120	12000
7	7	1	2016	336	336	168	2016	336	38304
8	2	3	3528	504	504	504	10584	1512	225792
9	3	2	5760	720	720	360	11520	1440	466560