Quotient modules isomorphic $ \Rightarrow$ submodules isomorphic

Question

Suppose $M$ is a module, $H$ and $K$ submodules.

If $$M/H \cong M/K$$ can I conclude that $H \cong K $ ?

If not, how to construct a counterexample ?

Answer 1

No, take for instance $M=\Bbb Z/2\Bbb Z\oplus \Bbb Z$ and $H=\Bbb Z$, $K=\Bbb Z/2\Bbb Z\oplus 2\Bbb Z$. In both cases the quotient is $\Bbb Z/2\Bbb Z$. You can even turn this into an examps with finite groups, by considering $M'=\Bbb Z/2\Bbb Z\oplus \Bbb Z/4\Bbb Z$, $H=\Bbb Z/4\Bbb Z$ and $K=\Bbb Z/2\Bbb Z\oplus 2\Bbb Z/4\Bbb Z$ instead.

Answer 2

The answer is no, here's the counterexample.

Let $M=\mathbb Z_{4} \times \mathbb Z_{2}$ these are $\mathbb Z$-modules. The modules $H=\mathbb Z_{4} \times \{0\}$ and $K=2\mathbb Z_{4} \times \mathbb Z_{2}$ and $M/H \cong M/K \cong \mathbb Z_{2}$ but clearly $\mathbb Z_{4} \times \{0\}$ and $2\mathbb Z_{4} \times \mathbb Z_{2} \cong \mathbb Z_{2} \times \mathbb Z_{2}$ are clearly not isomorphic.