I am not sure how to start solving this question for my modern algebra course. I understand Lagrange's theorem, but I am not sure if I should use it here. I just need help on where to start.
Let G and H be cyclic groups of orders 7, m respectively. Let us write eG as the identity of G and eH as the identity of H. Assume that G =< a > and H =< b >. Let f : G → H be a homomorphism such that f(a) = b^r where r > 0 is an integer. Answer the following: If m = 25 show that r must be 0. Explain why this means f(x) = eH for all x ∈ G.
For all $x\in G$, we have that $(f(x))^n=f(x^n)=e_H$ whenever $x^n=e_G$. Therefore, the order of $f(x)$ divides the order of $x$.
Now, what are the possible orders of elements in $G$ and $H$? Can we ever map anything to an element that doesn't have order $1$?