Element structure of special linear group:SL(2,R)

This article aims to discuss the element structure of special linear group:SL(2,R).

Related information

 * Element structure of special linear group of degree two over a field: This is very general, but requres a large number of cases and takes a lot of effort to understand in its entirety.
 * Element structure of special linear group of degree two over a finite field

Conjugacy class structure
To deduce this from element structure of special linear group of degree two over a field, we need to use the following facts, about $$\R$$, the field of real numbers:


 * The group $$\R^*/(\R^*)^2$$ is cyclic of order two, with representatives $$1, -1 $$.
 * The only separable quadratic extension of $$\R$$ is the field of complex numbers, obtained by adjoining a square root of -1.
 * Further, the algebraic norm of any nonzero complex number is a positive real number, and in particular, it is a square. Thus, $$\R^*/N(\mathbb{C}^*)$$ has size two.