Isomorphisms between finite abelian groups and cyclic groups

Question

If G is abelian of order 175 and H is cyclic of order 25 and there is a homomorphism from G onto H then what is G isomorphic to?

I can see how G is isomorphic to either $C_{25} * C_7$ or to $C_5 * C_5 * C_7$. Can there be a homomorphism from $C_5 * C_5$ onto H or am I restricted to the $G=C_{25} * C_7$ case (as clearly H and $C_{25}$ are isomorphic)?

Answer 1

Note that we always have that $|\varphi(x)|\mid |x|$ (the order of the image divides the order of the initial value. If you had that $\varphi$ was onto, then there would have an $x\in G$ such that $\varphi(x)$ generates $C_{25}$, so we would have $25\mid |x|$. Are there any elements in $C_5*C_5*C_7$ of order divisible by $25$?