Order formulas for linear groups of degree two

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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
3 [SHOW MORE]
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