Let $C = \{z \in \mathbb C \mid |z| = 1\}.$ Let $f_\theta : \mathbb C \to \mathbb C$ be given by $f_\theta (z) = e^{i\theta z}$. Let $F = \{f_\theta | \theta \in \mathbb R\}$. Consider the set $F$ under the operation of composition of functions $\circ$.
(a) Is $(F, \circ)$ a semigroup? Justify your answer.
(b) Is $(F, \circ )$ a monoid ? Justify your answer.
(c) Is $(F, \circ )$ a group? Justify your answer.
(d) Is the map $\psi : (\mathbb R, +) \to (F, \circ )$ given by $\psi(\theta ) = f_\theta $ a homomorphism? Justify your answer.
(e) Is that map $\psi$ from part (d) an isomorphism from $(\mathbb R, +)\to (F, \circ )$? Justify your answer.
I understand the terms, but am confused in relation to the function and relating the terms to the function
Notice that the map $f_\theta$ is just rotating $\mathbb{C}$ anticlockwise by an angle $\theta$. Since the definition of function composition means that it is always associative, this should be enough for you to do (a), (b), (c), and (d).
Here's a hint for (e) by way of analogy: on the set $\{-1,1\}$, the functions $x \mapsto |x|$ and $x\mapsto x^2$ are equal, even though they have been written differently. Can something similar happen in your case?