Find the number of generators for $(\mu_{25}, .)$
I know that $\bar{a} \in \mathbb{Z}/n \mathbb{Z}$ is a generator iff the $\gcd(a, n)=1$. So as I understand there is 20 generators. I know that $\mathbb{Z}/n \mathbb{Z} \cong (\mu_{25}, .)$.
Will I have 20 generators as well for $\mu_{25}$? In other words, I would like to know if the number of generator are preserved between $(\mu_{25}, .)$ and $\mathbb{Z}/n \mathbb{Z}$.
Definition: $\mu_n = \left \{ e^{\frac{2 k \pi i}{n}} : 1 \leq k \leq 24 \right\}$
Yes, you will have $\varphi(25)=20$ generators. Isomorphisms preserve both the orders of the groups and the orders of their elements. Also, generators are mapped onto generators; see here and here.