Kernels and homomorphisms

Question

Let us have two groups $A, B$ and a homomorphism $\theta: A \longrightarrow B$.

Let $|A|=xy,|B|=yz$ where $x,y,z$ are distinct primes.

If we have a subgroup $C$ of $A$ and $|C|=x$ how do we show that $C$ is a subset of $ker(\theta)$?

Answer 1

The question is totally different now. I answered the original question (click on "edited" to see the first version).

The order of the image of $C$ must (by Lagrange) divide $x$ and $yz$. The only such natural number is 1. So the image is $\{1\}$ and $C$ is inside the kernel.