Element structure of general linear group of degree two over a finite field
This article gives specific information, namely, element structure, about a family of groups, namely: general linear group of degree two.
View element structure of group families | View other specific information about general linear group of degree two
This article gives information on the element structure for a finite field. If you are interested in rings such asor
, or Galois rings, see element structure of general linear group of degree two over a finite discrete valuation ring.
This article gives the element structure of the general linear group of degree two over a finite field. Similar structure works over an infinite field or a field of infinite characteristic, with suitable reinterpretation and modification.
We denote the order (or size) of the field by and the characteristic of the field by
.
is a power of
.
Related information
Generalizing to infinite fields and non-fields
- Element structure of general linear group of degree two over a field
- Element structure of general linear group of degree two over a finite discrete valuation ring
Related groups over finite fields
- Element structure of special linear group of degree two over a finite field
- Element structure of projective general linear group of degree two over a finite field
- Element structure of projective special linear group of degree two over a finite field
Summary
Item | Value |
---|---|
order | ![]() |
exponent | ![]() |
number of conjugacy classes | ![]() |
Particular cases
![]() |
![]() |
general linear group ![]() |
order of the group (= ![]() |
number of conjugacy classes (= ![]() |
element structure page |
---|---|---|---|---|---|
2 | 2 | symmetric group:S3 | 6 | 3 | element structure of symmetric group:S3 |
3 | 3 | general linear group:GL(2,3) | 48 | 8 | element structure of general linear group:GL(2,3) |
4 | 2 | direct product of A5 and Z3 | 180 | 15 | |
5 | 5 | general linear group:GL(2,5) | 480 | 24 | element structure of general linear group:GL(2,5) |
7 | 7 | general linear group:GL(2,7) | 2016 | 48 | element structure of general linear group:GL(2,7) |
Conjugacy class structure
There is a total of elements, and there are
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 ![]() |
---|---|---|---|---|---|---|---|---|
Diagonalizable over ![]() |
![]() ![]() |
![]() ![]() |
![]() ![]() |
1 | ![]() |
![]() |
Yes | Yes |
Diagonalizable over ![]() ![]() |
Pair of conjugate elements of ![]() |
![]() |
Same as characteristic polynomial | ![]() |
![]() |
![]() |
Yes | No |
Not diagonal, has Jordan block of size two | ![]() ![]() |
![]() ![]() |
Same as characteristic polynomial | ![]() |
![]() |
![]() |
No | No |
Diagonalizable over ![]() |
![]() ![]() |
![]() |
Same as characteristic polynomial | ![]() |
![]() |
![]() |
Yes | Yes |
Total | NA | NA | NA | NA | ![]() |
![]() |
Order information
Information of elements of an exact order
Type of element | Case of order ![]() ![]() |
Case of order ![]() ![]() ![]() |
Case of order ![]() ![]() ![]() |
---|---|---|---|
Central, diagonal over ![]() |
![]() ![]() |
0, 0 | 0, 0 |
Diagonalizable over ![]() ![]() |
0 conjugacy classes, 0 elements | ![]() ![]() |
0 conjugacy classes, 0 elements |
Diagonalizable over ![]() |
![]() ![]() |
0 conjugacy classes, 0 elements | 0 conjugacy classes, 0 elements |
Jordan block of size two | 0 conjugacy classes, 0 elements | 0 conjugacy classes, 0 elements | ![]() ![]() |
Total | ![]() ![]() |
![]() ![]() |
![]() ![]() |
Central elements
The center is a subgroup of order , and its elements are diagonal matrices of the form:
The subgroup is isomorphic to the multiplicative group of the field of elements, and since multiplicative group of finite field is cyclic, it is a cyclic group of order
.
Real and rational conjugacy
We have the following:
Item | Value |
---|---|
Number of equivalence classes under real conjugacy in the whole group for elements in the center | ![]() ![]() ![]() ![]() |
Number of real elements in the center | 2 if ![]() ![]() |
Number of equivalence classes under rational conjugacy in the whole group for elements in the center | ![]() ![]() |
Number of rational elements in the center | 2 if ![]() ![]() |
Order information
For a given dividing
:
Item | Value |
---|---|
Number of elements of order precisely ![]() |
![]() |
Number of elements of order dividing ![]() |
![]() |
Elements diagonalizable over
but not 
These elements have pairs of distinct eigenvalues over that are conjugate over
. The unique non-identity automorphism of
over
is the map
, so these two elements are
powers of each other, i.e., if one of them is
, the other one is
.
The conjugacy class is parameterized by the pair . Here is more information:
Item | Value | Explanation |
---|---|---|
Trace of conjugacy class | ![]() ![]() |
Trace is sum of eigenvalues |
Norm or determinant of conjugacy class | ![]() ![]() |
Product of eigenvalues |
Minimal polynomial of conjugacy class | ![]() ![]() ![]() |
Distinct eigenvalues |
Characteristic polynomial of conjugacy class | Same as minimal polynomial | Distinct eigenvalues |
Centralizer of any element of conjugacy class | It is isomorphic to the multiplicative group ![]() ![]() |
Any choice of element gives a particular interpretaiton of a two-dimensional vector space over ![]() ![]() ![]() |
Size of conjugacy class | ![]() |
It equals the index of the centralizer |
Here is information on the collection of all such conjugacy classes:
Item | Value | Explanation |
---|---|---|
number of conjugacy classes of this type | ![]() |
First explanation: The conjugacy classes are classified by pairs ![]() ![]() ![]() ![]() ![]() Second explanation: The conjugacy classes are characterized by irreducible monic quadratic polynomials. The total number of monic quadratics is ![]() ![]() ![]() ![]() |
total number of elements of this type | ![]() |
Multiply the size of each conjugacy class with the number of conjugacy classes of this type |
number of centralizers, or equivalently, number of different ways of realizing ![]() |
![]() |
Order information
A number occurs as the order of an element in one of these conjugacy classes if
but
does not divide
. More information below:
Count | Number of conjugacy classes | Number of elements (obtained by multiplying number of conjugacy classes by size of each conjugacy class, which is ![]() |
---|---|---|
Elements of order precisely ![]() ![]() ![]() ![]() |
![]() ![]() |
![]() |
Elements of order dividing ![]() |
? |
Elements diagonalizable over
with distinct diagonal entries
Note that we are including here only invertible elements, hence both eigenvalues must be in .
Each such conjugacy class is specified by an unordered pair of distinct elements of , say
.
Item | Value | Explanation |
---|---|---|
Trace of conjugacy class | ![]() ![]() |
Trace is sum of eigenvalues |
Norm or determinant of conjugacy class | ![]() ![]() |
Product of eigenvalues |
Minimal polynomial of conjugacy class | ![]() ![]() ![]() |
Distinct eigenvalues |
Characteristic polynomial of conjugacy class | Same as minimal polynomial | Distinct eigenvalues |
Centralizer of any element of conjugacy class | It is isomorphic to ![]() ![]() |
This is easiest to see for the representative that is diagonal: its centralizer is the subgroup of diagonal matrices. |
Size of conjugacy class | ![]() |
It equals the index of the centralizer |
Here is combined information for all conjugacy classes:
Item | Value | Explanation |
---|---|---|
number of conjugacy classes of this type | ![]() |
We need to pick for the eigenvalues two distinct elements ![]() ![]() ![]() |
total number of elements of this type | ![]() |
Multiply number of conjugacy classes and size of each conjugacy class. |
number of centralizers | ![]() |
Order information
A natural number occurs as the order of an element in one of these conjugacy classes if and only if
divides
but is greater than
. For such a value of
, we have:
Count | Number of conjugacy classes | Number of elements (obtained by multiplying number of conjugacy classes with size of each conjugacy class) |
---|---|---|
order precisely ![]() |
![]() ![]() |
![]() |
order dividing ![]() |
![]() |
![]() |
Elements with Jordan block of size two
These are elements conjugate to elements of the form:
Each conjugacy class is parameterized by the value of .
Item | Value | Explanation |
---|---|---|
Trace of conjugacy class | ![]() |
|
Norm or determinant of conjugacy class | ![]() |
|
Minimal polynomial of conjugacy class | ![]() |
Repeated eigenvalue |
Characteristic polynomial of conjugacy class | ![]() |
|
Centralizer of any element in conjugacy class | Isomorphic to direct product of additive group of ![]() ![]() |
For the element ![]() ![]() ![]() ![]() |
Size of conjugacy class | ![]() |
size of conjugacy class equals index of centralizer |
Here is the information on the collection of all conjugacy classes:
Item | Value | Explanation |
---|---|---|
number of conjugacy classes of this type | ![]() |
The parameter varies over ![]() |
total number of elements of this type | ![]() |
multiply number of conjugacy classes with size of each conjugacy class |
number of centralizers | ![]() |
First, note that all centralizers can be attained from just one conjugacy class. Next, note that ![]() ![]() |
Order information
The possible orders are , where
divides
and
is the underlying prime (the characteristic) whose power is
. For each order
, the number of conjugacy classes is
, the Euler totient function.