# How could I describe homomorphisms from $C_m \rightarrow C_n$?

E.g.: If $m = 21$ and $n = 15$. I realise I could enumerate the number of bijective mappings but I suspect I would be excluding other homomorphisms. How would I best approach this problem?

• Well, you just have to say what the generator maps to. In the case you mentioned, what are the possible orders of the image of a generator? – lulu Aug 8 '17 at 20:08
• I see, would that be 15, 5, 3 or 1? – silverjoe Aug 8 '17 at 20:19
• How did you get those answers? $1$ has order $15$, for example...how could the generator of a group of order $21$ map to an element of order $15$? – lulu Aug 8 '17 at 20:21
• That's true, what I did is I assumed that homorphisms would map the generator $x$ of $C_21$, to some power positive integer power $m$, then I used the formula $ord(x^m) = n/gcd(m, n)$ to determine possible order n in $C_15$. But that is true it doesn't make sense for an order $21$ element to map to one of order $15$. – silverjoe Aug 8 '17 at 20:34
• Indeed, if $\sigma$ is the generator of the group of order $21$, and $\phi$ the homomorphism, then we know that $\phi(\sigma)^{15}=e=\phi(\sigma)^{21}$ so the order of $\phi(\sigma)$ has to be a divisor of $\gcd(15,21)=3$. – lulu Aug 8 '17 at 20:36