# 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 $r = \log_pq$, 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]
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]
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]
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
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]
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]
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]
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]

### 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