Is there any group $G$ with $ord(G)=20$ so that $\varphi:G\rightarrow \mathbb{Z}_{15}$ is epimorphism?

Question

Is there any group $G$ with $ord(G)=20$ so that $\varphi:G\rightarrow \mathbb{Z}_{15}$ is epimorphism?

I thout about $\mathbb{Z}_{20}$ to map $\varphi:\mathbb{Z}_{20}\rightarrow\mathbb{Z}_{15}$,such that $\varphi$ defined by $\varphi(a)=a\mod 15$

but $\varphi$ is not homomorphusm because:

$\varphi(ab)=(a+b)\mod 15$

$\varphi(a)\varphi(b)=a \mod 15 +b \mod 15\not=\varphi(ab)$

Answer 1

Hint: for any homomorphism $\varphi\colon G\to\mathbb{Z}_{15}$ there is an induced injective homomorphism $\hat{\varphi}\colon G/\ker\varphi\to\mathbb{Z}_{15}$ that has the same image as $\varphi$. If $\varphi$ is surjective, then $|G/\ker\varphi|=15$. Now you should be able to conclude by recalling that $$ |G/\ker\varphi|=\frac{|G|}{|\ker\varphi|} $$