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.
Here, denotes the order of the finite field and the group we work with is .
|order||options for first row, options for second row.|
|exponent||There is an element of order and an element of order .|
|number of conjugacy classes||There are conjugacy classes of semisimple matrices and conjugacy classes of unipotent matrices.|
|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.|