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.
Infinite rings and fields
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.
|| options for first row, options for second row.
|| 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.
| Inner automorphism group
|| Projective general linear group of degree two
|| Quotient by the center, which is the group of scalar matrices.
|| 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.