General linear group of degree two

Definition

For a unital ring

The general linear group of degree two over a unital ring $R$ is defined as the group, under matrix multiplication, of invertible $2\times 2$ matrices with entries in $R$. It is denoted $GL(2,R)$.

For a commutative unital ring

When $R$ is a commutative unital ring, a $2\times 2$ matrix over $R$ being invertible is equivalent to its determinant being an invertible element of $R$, so the general linear group $GL(2,R)$ is defined as the following group of matrices under matrix multiplication:

$GL(2,R):=\{\left(\begin{array}{cc}a& b\\ c& d\\ \end{array}\right)\mid a,b,c,d\in R,ad-bc\text{ is an invertible element of }R\}$

For a field

For a field $K$, an element is invertible iff it is nonzero, so the general linear group $GL(2,K)$ is defined as the following group of matrices under matrix multiplication:

$GL(2,K):=\{\left(\begin{array}{cc}a& b\\ c& d\\ \end{array}\right)\mid a,b,c,d\in K,ad-bc\ne 0\}$

For a prime power

For a prime power $q$, $GL(2,q)$ or $G{L}_2(q)$ denotes the general linear group of degree two over the finite field (unique up to isomorphism) with $q$ elements. This is a field of characteristic $p$, where $p$ is the prime number whose power is $q$.

Particular cases

Finite fields

Common name for general linear group of degree two	Field	Size of field	Order of group
symmetric group:S3	field:F2	2	6
general linear group:GL(2,3)	field:F3	3	48
direct product of A5 and Z3	field:F4	4	180
general linear group:GL(2,5)	field:F5	5	480

Infinite rings and fields

Name of ring/field	Common name for general linear group of degree two
Ring of integers $\mathbb{Z}$	general linear group:GL(2,Z)
Field of rational numbers $\mathbb{Q}$	general linear group:GL(2,Q)
Field of real numbers $\mathbb{R}$	general linear group:GL(2,R)
Field of complex numbers $\mathbb{C}$	general linear group:GL(2,C)

Arithmetic functions

Here, $q$ denotes the order of the finite field and the group we work with is $GL(2,q)$. $p$ is the characteristic of the field, i.e., it is the prime whose power $q$ is.

Function	Value	Explanation
order	$(q^2-1)(q^2-q)=q^4-q^3-q^2+q=q(q+1)(q-1{)}^2$	$q^2-1$ options for first row, $q^2-q$ options for second row. See order formulas for linear groups of degree two
exponent	$p(q^2-1)=p(q-1)(q+1)$	There is an element of order $q^2-1$ and an element of order $p(q-1)$. All elements have order dividing $p(q-1)$ or $q^2-1$.
number of conjugacy classes	$q^2-1=(q+1)(q-1)$	There are $q(q-1)$ conjugacy classes of semisimple matrices and $q-1$ conjugacy classes of matrices with repeated eigenvalues.

Group properties

Property	Satisfied?	Explanation
abelian group	No	The matrices $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$ and $\left(\begin{array}{cc}0& 1\\ 1& 0\\ \end{array}\right)$ don't commute.
nilpotent group	No	$PSL(2,q)$ is simple for $q\ge 4$, and we can check the cases $q=2,3$ separately.
solvable group	Yes if $q=2,3$, no otherwise.	$PSL(2,q)$ is simple for $q\ge 4$.
supersolvable group	Yes if $q=2$, no otherwise.	$PSL(2,q)$ is simple for $q\ge 4$, and we can check the cases $q=2,3$ separately.

Elements

Further information: Element structure of general linear group of degree two over a finite field

There is a total of $q(q+1)(q-1{)}^2=q^4-q^3-q^2+q$ elements, and there are $q^2-1=(q-1)(q+1)$ conjugacy classes of elements.

For background on how this conjugacy class structure can be obtained and also generalized to general linear groups of degree three or more, refer to conjugacy class size formula in general linear group over finite field.

Nature of conjugacy class	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Semisimple?	Diagonalizable over ${\mathbb{F}}_q$?
Diagonalizable over ${\mathbb{F}}_q$ with equal diagonal entries, hence a scalar	$\{a,a\}$ where $a\in {\mathbb{F}}_q^{\ast }$	$(x-a{)}^2$ where $a\in {\mathbb{F}}_q^{\ast }$	$x-a$ where $a\in {\mathbb{F}}_q^{\ast }$	1	$q-1$	$q-1$	Yes	Yes
Diagonalizable over ${\mathbb{F}}_{q^2}$, not over ${\mathbb{F}}_q$. Must necessarily have no repeated eigenvalues.	Pair of conjugate elements of ${\mathbb{F}}_{q^2}$	$x^2-ax+b$, irreducible	Same as characteristic polynomial	$q(q-1)$	$\frac{q(q-1)}{2}=\frac{q^2-q}{2}$	$q^2(q-1{)}^2/2=(q^4-2q^3+q^2)/2$	Yes	No
Not diagonal, has Jordan block of size two	$a$ (multiplicity two) where $a\in {\mathbb{F}}_q^{\ast }$	$(x-a{)}^2$ where $a\in {\mathbb{F}}_q^{\ast }$	Same as characteristic polynomial	$q^2-1$	$q-1$	$(q^2-1)(q-1)=q^3-q^2-q+1$	No	No
Diagonalizable over ${\mathbb{F}}_q$ with distinct diagonal entries	$\lambda ,\mu$ (interchangeable) distinct elements of ${\mathbb{F}}_q^{\ast }$	$x^2-(\lambda +\mu )x+\lambda \mu$	Same as characteristic polynomial	$q(q+1)$	$\frac{(q-1)(q-2)}{2}=\frac{q^2-3q+2}{2}$	$\frac{q(q+1)(q-1)(q-2)}{2}=\frac{q^4-2q^3-q^2+2q}{2}$	Yes	Yes
Total	NA	NA	NA	NA	$q^2-1$	$q(q+1)(q-1{)}^2=q^4-q^3-q^2+q$

Subgroup-defining functions

Subgroup-defining function	Value	Explanation
Center	The subgroup of scalar matrices. Cyclic of order $q-1$	Center of general linear group is group of scalar matrices over center.
Commutator subgroup	Except the case of $GL(2,2)$, it is the special linear group of degree two, which has index $q-1$.	Commutator subgroup of general linear group is special linear group

Quotient-defining functions

Subgroup-defining function	Value	Explanation
Inner automorphism group	Projective general linear group of degree two	Quotient by the center, which is the group of scalar matrices.
Abelianization	This is isomorphic to the multiplicative group of the field.	Quotient by the commutator subgroup, which is the special linear group, which is the kernel of the determinant map that surjects to the multiplicative group of the field.