Let C be the group of continuous bijections on the unit interval to itself using functional composition as the group operation. Let P be the set of all the polynomials in C.
i) Show that P is a group under the operation ii) Show that P is not a subgroup of C iii) Show there are only two first degree polynomials in P and list their formulas iv) Show there are infinitely many quadratic polynomials in P
First of all, if P is closed under the group operation how can it not be a subgroup of C?
For iii) Clearly f(x)=x is one, but I'm lost on the other. I think f(x)=cos(x) or f(x)=sin(x) has promise but it clearly cannot be both...
iv) I don't even know where to begin.