General linear group of degree two: Difference between revisions

Latest revision as of 21:13, 18 September 2012

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 $G L (2, R)$ .

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 $G L (2, R)$ is defined as the following group of matrices under matrix multiplication:

$G L (2, R) : = {(\begin{matrix} a & b \\ c & d \end{matrix}) ∣ a, b, c, d \in R, a d - b c is an invertible element of R}$

For a field

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

$G L (2, K) : = {(\begin{matrix} a & b \\ c & d \end{matrix}) ∣ a, b, c, d \in K, a d - b c \neq 0}$

For a prime power

For a prime power $q$ , $G L (2, q)$ 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 $Z$	general linear group:GL(2,Z)
Field of rational numbers $Q$	general linear group:GL(2,Q)
Field of real numbers $R$	general linear group:GL(2,R)
Field of complex numbers $C$	general linear group:GL(2,C)

Arithmetic functions

Here, $q$ denotes the order of the finite field and the group we work with is $G L (2, q)$ . $p$ 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 $(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ and $(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$ don't commute.
nilpotent group	No	$P S L (2, q)$ is simple for $q \geq 4$ , and we can check the cases $q = 2, 3$ separately.
solvable group	Yes if $q = 2, 3$ , no otherwise.	$P S L (2, q)$ is simple for $q \geq 4$ .
supersolvable group	Yes if $q = 2$ , no otherwise.	$P S L (2, q)$ is simple for $q \geq 4$ , and we can check the cases $q = 2, 3$ separately.

Elements

Information based on ring type

Ring type	Element structure page
field	element structure of general linear group of degree two over a field
finite field	element structure of general linear group of degree two over a finite field
finite discrete valuation ring	element structure of general linear group of degree two over a finite discrete valuation ring
division ring	element structure of general linear group of degree two over a division ring

Conjugacy class structure (case of a field)

Nature of conjugacy class	Eigenvalues	Characteristic polynomial	Minimal polynomial	What set can each conjugacy class be identified with? (rough measure of size of conjugacy class)	What can the set of conjugacy classes be identified with (rough measure of number of conjugacy classes)	What can the union of conjugacy classes be identified with?	Semisimple?	Diagonalizable over $K$ ?
Diagonalizable over $K$ with equal diagonal entries, hence a scalar	${a, a}$ where $a \in K^{*}$	$(x - a)^{2}$ where $a \in K^{*}$	$x - a$ where $a \in K^{*}$	one-point set	$K^{*}$	$K^{*}$	Yes	Yes
Diagonalizable over $K$ with distinct diagonal entries	$λ, μ$ (interchangeable) distinct elements of $K^{*}$	$x^{2} - (λ + μ) x + λ μ$	Same as characteristic polynomial	set of decompositions of a fixed two-dimensional vector space over $K$ as a direct sum of one-dimensional subspaces	the set $(\binom{K^{*}}{2})$	?	Yes	Yes
Diagonalizable over a quadratic extension of $K$ but not over $K$ itself. Must necessarily have no repeated eigenvalues.	Pair of conjugate elements of some separable quadratic extension of $K$	$x^{2} - a x + b$ , irreducible	Same as characteristic polynomial	?	?	?	Yes	No
Not diagonal, has Jordan block of size two with eigenvalue in $K$	$a$ (multiplicity two) where $a \in K^{*}$	$(x - a)^{2}$ where $a \in K^{*}$	Same as characteristic polynomial	?	?	?	No	No
Not diagonal, has Jordan block of size two with eigenvalue not in $K$	$a$ (multiplicity two) where $a$ is in a purely inseparable quadratic extension of $K$ , so $a^{2} \in K$ . This case arises only when $K$ has characteristic two and is not a perfect field	$x^{2} - t$ , $t$ a non-square in $K^{*}$ , $K$ has characteristic two	Same as characteristic polynomial	?	?	?	No	No

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 $G L (2, 2)$ , 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.

@@ Line 1: / Line 1: @@
+[[importance rank::2| ]]
 ==Definition==
-The '''general linear group of degree two''' over a [[field]] <math>k</math> (respectively, over a unital ring <math>R</math>), is defined as the group, under multiplication, of invertible <math>2 \times 2</math> matrices with entries in <math>k</math>. It is denoted <math>GL(2,k)</math> (respectively, <math>GL(2,R)</math>).
+===For a unital ring===
-For a prime power <math>q</math>, <math>GL(2,q)</math> or <math>GL_2(q)</math> denotes the general linear group of degree two over the field (unique up to isomorphism) with <math>q</math> elements.
+The '''general linear group of degree two''' over a unital ring <math>R</math> is defined as the group, under matrix multiplication, of invertible <math>2 \times 2</math> matrices with entries in <math>R</math>. It is denoted <math>GL(2,R)</math>.
+===For a commutative unital ring===
+When <math>R</math> is a commutative unital ring, a <math>2 \times 2</math> matrix over <math>R</math> being invertible is equivalent to its determinant being an invertible element of <math>R</math>, so the general linear group <math>GL(2,R)</math> is defined as the following group of matrices under matrix multiplication:
+<math>GL(2,R) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in R, ad - bc \mbox{ is an invertible element of } R \right \}</math>
+===For a field===
+For a [[field]] <math>K</math>, an element is invertible iff it is nonzero, so the general linear group <math>GL(2,K)</math> is defined as the following group of matrices under matrix multiplication:
+<math>GL(2,K) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in K, ad - bc \ne 0 \right \}</math>
+===For a prime power===
+For a [[prime power]] <math>q</math>, <math>GL(2,q)</math> or <math>GL_2(q)</math> denotes the general linear group of degree two over the [[finite field]] (unique up to isomorphism) with <math>q</math> elements. This is a field of characteristic <math>p</math>, where <math>p</math> is the [[prime number]] whose power is <math>q</math>.
 ==Particular cases==
@@ Line 39: / Line 56: @@
 Here, <math>q</math> denotes the order of the finite field and the group we work with is <math>GL(2,q)</math>. <math>p</math> is the characteristic of the field, i.e., it is the prime whose power <math>q</math> is.
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
 ! Function !! Value !! Explanation
 |-
-| [[order of a group|order]] || <math>\! q^4 - q^3 - q^2  + q = q(q + 1)(q-1)^2</math> || <math>q^2 - 1</math> options for first row, <math>q^2 - q</math> options for second row.
+| [[order of a group|order]] || <math>\! (q^2 - 1)(q^2 - q) = q^4 - q^3 - q^2  + q = q(q + 1)(q-1)^2</math> || <math>q^2 - 1</math> options for first row, <math>q^2 - q</math> options for second row.<br>See [[order formulas for linear groups of degree two]]
 |-
 | [[exponent of a group|exponent]] || <math>\! p(q^2 - 1) = p(q-1)(q+1)</math> || There is an element of order <math>q^2 - 1</math> and an element of order <math>p(q - 1)</math>. All elements have order dividing <math>p(q - 1)</math> or <math>q^2 - 1</math>.
@@ Line 51: / Line 68: @@
 ==Group properties==
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
-!Property !! Satisfied !! Explanation
+!Property !! Satisfied? !! Explanation
 |-
-|[[Abelian group]] || No || The matrices <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}</math> don't commute.
+|[[abelian group]] || No || The matrices <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}</math> don't commute.
 |-
-|[[Nilpotent group]] || No || <math>PSL(2,q)</math> [[projective special linear group is simple|is simple]] for <math>q \ge 4</math>, and we can check the cases <math>q = 2, 3</math> separately.
+|[[nilpotent group]] || No || <math>PSL(2,q)</math> [[projective special linear group is simple|is simple]] for <math>q \ge 4</math>, and we can check the cases <math>q = 2, 3</math> separately.
 |-
-|[[Solvable group]] || Yes if <math>q = 2,3</math>, no otherwise. || <math>PSL(2,q)</math> [[projective special linear group is simple|is simple]] for <math>q \ge 4</math>.
+|[[solvable group]] || Yes if <math>q = 2,3</math>, no otherwise. || <math>PSL(2,q)</math> [[projective special linear group is simple|is simple]] for <math>q \ge 4</math>.
 |-
-|[[Supersolvable group]] || Yes if <math>q - 2</math>, no otherwise. || <math>PSL(2,q)</math> is simple for <math>q \ge 4</math>, and we can check the cases <math>q = 2, 3</math> separately.
+|[[supersolvable group]] || Yes if <math>q = 2</math>, no otherwise. || <math>PSL(2,q)</math> is simple for <math>q \ge 4</math>, and we can check the cases <math>q = 2, 3</math> separately.
 |}
 ==Elements==
-{{further|[[Element structure of general linear groups of degree two]]}}
+===Information based on ring type===
-The elements are as follows:
+{| class="sortable" border="1"
+! Ring type !! Element structure page
-# There are <math>q - 1</math> conjugacy classes of size one, corresponding to the central elements.
+|-
-# There are <math>q(q-1)/2</math> conjugacy classes of size <math>q(q-1)</math> each. These are obtained as follows: we know that there is a field extension of size <math>q^2</math>, which can be identified with a vector space of dimension two over the field of <math>q</math> elements. Left multiplication by an element in the field gives a <math>2 \times 2</math> matrix. There are <math>q(q-1) = q^2 - q</math> of these elements corresponding to a particular choice of basis for the field that are ''not'' scalar matrices. These come in pairs of conjugate elements (conjugate in the sense of field extensions) that are hence also conjugate in <math>GL(2,q)</math>. Thus, there are <math>q(q-1)/2</math> conjugacy classes here. Further, for each such element, the centralizer is the multiplicative group of the field with <math>q^2</math> elements, which has order <math>q^2 - 1</math>. The quotient has order <math>q(q+1)(q-1)^2/(q^2 - 1) = q(q-1)</math>.
+| [[field]] || [[element structure of general linear group of degree two over a field]]
-# There are <math>(q-1)(q-2)/2</math> conjugacy classes of size <math>q(q+1)</math> each. These correspond to elements that are diagonalizable with distinct eigenvalues. The number <math>(q-1)(q-2)/2</math> corresponds to choices of two distinct elements among the <math>q - 1</math>. Further, the centralizer of a diagonal matrix is the group of diagonal matrices, and has order <math>(q-1)^2</math>. The quotient has order <math>q(q+1)(q-1)^2/(q-1)^2 = q(q+1)</math>.
+|-
-# There are <math>q-1</math> conjugacy classes of size <math>(q-1)(q+1)</math> each.
+| [[finite field]] || [[element structure of general linear group of degree two over a finite field]]
+|-
+| [[finite discrete valuation ring]] || [[element structure of general linear group of degree two over a finite discrete valuation ring]]
+|-
+| [[division ring]] || [[element structure of general linear group of degree two over a division ring]]
+|}
+===Conjugacy class structure (case of a field)===
+{{#lst:element structure of general linear group of degree two over a field|conjugacy class structure}}
 ==Subgroup-defining functions==