I am currently using "A first course in abstract algebra" by Fraleigh for self-study.
Exercise 3.13 asks whether or not $\langle F, + \rangle$ with $\langle F, + \rangle$ is an isomorphism under the function $\phi(f)(x) = \int_{0}^{x} f(t) dt$.
The text describes 3 steps to check for an isomorphism regarding 2 binary structures $\langle S, * \rangle$ and $\langle S', *' \rangle$:
- $\phi$ is one-to-one: $\phi(x) = \phi(y) \implies x=y$.
- $\phi$ is an onto function from the domain to the codomain: $s' \in S $, there exists $s \in S$ such that $\phi(s) = s'$.
- Finally, $\phi(x*y) = \phi(x) *' \phi(y)$.
Regarding exercise 3.13, I verified that $\phi$ is one-to-one (correct me if I'm wrong) but I'm stuck on how to verify the next 2 steps (if they even can be verified).
Any help would be appreciated!