TITLE: Do subgroup and quotient group define a group?
QUESTION [10 upvotes]: Does a (normal) subgroup along with its corresponding quotient group define a group completely? Or are there groups with isomorphic normal subgroups and isomorphic corresponding quotient groups which are themselves as a whole not isomorphic?

REPLY [5 votes]: No, your question is concerned with "Extension theory for groups". If you have $N \unlhd G$ a normal subgroup then let $\phi$ be the natural inclusion (injective) $\phi \colon N \hookrightarrow G$ and $N$ is the kernel of the natural map $\pi \colon G \twoheadrightarrow G / N$. Note that $im(\phi)=\phi(N)=N=ker(\pi)$. That is to say that $G$ fits into the sequence
$$ 0 \rightarrow N \rightarrow G \rightarrow G /N \rightarrow 0$$
which is exact, i.e. the second arrow is injective (this is $\phi$), the third is surjective (this is $\pi$) and $im(\phi)=ker(\pi)$. One now asks which groups $G$ fit into such an exact sequence, i.e for given groups $\Gamma$ and $Q$:
$$ 0 \rightarrow \Gamma \hookrightarrow G \twoheadrightarrow Q \rightarrow 0.$$
This problem is concerned with cohomology theory of groups (see the corresponding wikipedia article and probably many books with subject "groups")