General linear group of degree two
The general linear group of degree two over a field (respectively, over a unital ring ), is defined as the group, under multiplication, of invertible matrices with entries in . It is denoted (respectively, ).
For a prime power , or denotes the general linear group of degree two over the field (unique up to isomorphism) with elements.
|Size of field||Common name for general linear group of degree two|
|general linear group:GL(2,3)|
|general linear group:GL(2,4)|
|general linear group:GL(2,5)|
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)|
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.
|order||options for first row, options for second row.|
|exponent||There is an element of order and an element of order . All elements have order dividing or .|
|number of conjugacy classes||There are conjugacy classes of semisimple matrices and conjugacy classes of matrices with repeated eigenvalues.|
|Abelian group||No||The matrices and don't commute.|
|Nilpotent group||No||is simple for , and we can check the cases separately.|
|Solvable group||Yes if , no otherwise.||is simple for .|
|Supersolvable group||Yes if , no otherwise.||is simple for , and we can check the cases separately.|
Further information: Element structure of general linear groups of degree two
The elements are as follows:
- There are conjugacy classes of size one, corresponding to the central elements.
- There are conjugacy classes of size each. These are obtained as follows: we know that there is a field extension of size , which can be identified with a vector space of dimension two over the field of elements. Left multiplication by an element in the field gives a matrix. There are 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 . Thus, there are conjugacy classes here. Further, for each such element, the centralizer is the multiplicative group of the field with elements, which has order . The quotient has order .
- There are conjugacy classes of size each. These correspond to elements that are diagonalizable with distinct eigenvalues. The number corresponds to choices of two distinct elements among the . Further, the centralizer of a diagonal matrix is the group of diagonal matrices, and has order . The quotient has order .
- There are conjugacy classes of size each.
|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 , it is the special linear group of degree two, which has index .||Commutator subgroup of general linear group is special linear group|
|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.|