General linear group of degree two
Contents
Definition
For a unital ring
The general linear group of degree two over a unital ring is defined as the group, under matrix multiplication, of invertible
matrices with entries in
. It is denoted
.
For a commutative unital ring
When is a commutative unital ring, a
matrix over
being invertible is equivalent to its determinant being an invertible element of
, so the general linear group
is defined as the following group of matrices under matrix multiplication:
For a field
For a field , an element is invertible iff it is nonzero, so the general linear group
is defined as the following group of matrices under matrix multiplication:
For a prime power
For a prime power ,
or
denotes the general linear group of degree two over the finite field (unique up to isomorphism) with
elements. This is a field of characteristic
, where
is the prime number whose power is
.
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 ![]() |
general linear group:GL(2,Z) |
Field of rational numbers ![]() |
general linear group:GL(2,Q) |
Field of real numbers ![]() |
general linear group:GL(2,R) |
Field of complex numbers ![]() |
general linear group:GL(2,C) |
Arithmetic functions
Here, denotes the order of the finite field and the group we work with is
.
is the characteristic of the field, i.e., it is the prime whose power
is.
Function | Value | Explanation |
---|---|---|
order | ![]() |
![]() ![]() See order formulas for linear groups of degree two |
exponent | ![]() |
There is an element of order ![]() ![]() ![]() ![]() |
number of conjugacy classes | ![]() |
There are ![]() ![]() |
Group properties
Property | Satisfied? | Explanation |
---|---|---|
abelian group | No | The matrices ![]() ![]() |
nilpotent group | No | ![]() ![]() ![]() |
solvable group | Yes if ![]() |
![]() ![]() |
supersolvable group | Yes if ![]() |
![]() ![]() ![]() |
Elements
Information based on ring type
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 ![]() |
---|---|---|---|---|---|---|---|---|
Diagonalizable over ![]() |
![]() ![]() |
![]() ![]() |
![]() ![]() |
one-point set | ![]() |
![]() |
Yes | Yes |
Diagonalizable over ![]() |
![]() ![]() |
![]() |
Same as characteristic polynomial | set of decompositions of a fixed two-dimensional vector space over ![]() |
the set ![]() |
? | Yes | Yes |
Diagonalizable over a quadratic extension of ![]() ![]() |
Pair of conjugate elements of some separable quadratic extension of ![]() |
![]() |
Same as characteristic polynomial | ? | ? | ? | Yes | No |
Not diagonal, has Jordan block of size two with eigenvalue in ![]() |
![]() ![]() |
![]() ![]() |
Same as characteristic polynomial | ? | ? | ? | No | No |
Not diagonal, has Jordan block of size two with eigenvalue not in ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
Same as characteristic polynomial | ? | ? | ? | No | No |
Subgroup-defining functions
Subgroup-defining function | Value | Explanation |
---|---|---|
Center | The subgroup of scalar matrices. Cyclic of order ![]() |
Center of general linear group is group of scalar matrices over center. |
Commutator subgroup | Except the case of ![]() ![]() |
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. |