/sci/ - Science & Math

>>12558939
forgot image

>Prove that [math]\alpha[/math] and [math]\beta[/math] are homomorphisms
but they aren't are they? unless I'm misunderstanding the problem, [math]\mathscr{L}(\mathbb{R})[/math] is the set of real functions, [math]f : \mathbb{R} \to \mathbb{R}[/math] such that [math]f^{-1}[/math] exists and the binary operation is function composition.
Now [eqn]
\alpha(f\cdot{g}) = (f \cdot{g}) (1) = f(g(1)) \neq f(1) g(1) = \alpha(f) \alpha (g)
[/eqn]