Is there any injective homomorphism (i.e. monomorphism) from a non-cyclic group of order $4$ to $\mathbb{Z}_8$?

Question

The only such possible group is $V$ (up to isomorphism). If $\phi$ be such an into homomorphism, then $o(\phi(V))=4$ and $\phi(V)$ being a subgroup of $\mathbb{Z}_8$, it must be cyclic with a generator of order $4$. But, $V$ has not element of order $4$. A contradiction.

Does this work?

Answer 1

That argument works. Given a monomorphism $f:G\to H$ between groups $G,H$, then $G$ is isomorphic to $f(G)$, and the two must therefore have the same number of elements of any given order.

Alternatively, there is only one element of order $2$ in $\Bbb Z_8$, while there are three in $V$.