Element structure of general linear group of degree two over a 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
- Element structure of general linear group of degree two over a finite field
- Element structure of general linear group of degree two over a finite discrete valuation ring
- Element structure of special linear group of degree two over a field
- Element structure of special linear group of degree two over a finite field
Conjugacy class structure
The very broad description of conjugacy class types given here is valid for all fields. However, in order to be more specific, we need to split into cases, which we do later in the article.
|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 with equal diagonal entries, hence a scalar||where||where||where||one-point set||Yes||Yes|
|Diagonalizable over with distinct diagonal entries||(interchangeable) distinct elements of||Same as characteristic polynomial||set of decompositions of a fixed two-dimensional vector space over as a direct sum of one-dimensional subspaces||the set||?||Yes||Yes|
|Diagonalizable over a quadratic extension of but not over itself. Must necessarily have no repeated eigenvalues.||Pair of conjugate elements of some separable quadratic extension of||, irreducible||Same as characteristic polynomial||?||?||?||Yes||No|
|Not diagonal, has Jordan block of size two with eigenvalue in||(multiplicity two) where||where||Same as characteristic polynomial||?||?||?||No||No|
|Not diagonal, has Jordan block of size two with eigenvalue not in||(multiplicity two) where is in a purely inseparable quadratic extension of , so . This case arises only when has characteristic two and is not a perfect field||, a non-square in , has characteristic two||Same as characteristic polynomial||?||?||?||No||No|
Identification between conjugacy classes and monic quadratic polynomials
The "characteristic polynomial" is a mapping:
Conjugacy classes in Monic quadratic polynomials over with nonzero constant term
The characteristic polynomial always has nonzero constant term because the constant term is the determinant, which must be invertible.
This mapping is surjective. Further, it is almost injective, with the following exception: for those monic quadratics that are of the form with , there are two conjugacy classes mapping to such a polynomial: the diagonalizable conjugacy class, and the non-diagonalizable conjugacy class, which corresponds to a Jordan block of size two.
Another way of putting this is that we have a bijection:
Conjugacy classes of non-central elements in Monic quadratic polynomials over with nonzero constant term
We know that the set of monic quadratics over with nonzero constant term can be identified with , with denoting the possibilities for the linear coefficient (negative of the trace) and the possibilities for the constant term (the determinant). Thus:
Conjugacy classes of non-central elements in
In total we can identify:
Conjugacy classes in
In particular, when is a finite field of size , this allows us to compute the total number of conjugacy classes to be .
In fact, with a little work, we can make the mapping nicer, and get:
Conjugacy classes in