Short exact sequence of groups

Definition
A short exact sequence of groups is an defining ingredient::exact sequence of groups with five terms, where the first and last term are the trivial group. Explicitly, it has the form:

$$1 \to N \to G \to Q \to 1$$

The exactness of the sequence is equivalent to three condition:


 * The homomorphism from $$N$$ to $$G$$ is injective, so that $$N$$ is isomorphic to its image, which is a subgroup of $$G$$. We often abuse notation by conflating $$N$$ with its image in $$G$$.
 * The homomorphism from $$G$$ to $$Q$$ is surjective, so that $$Q$$ is isomorphic to a quotient group of $$G$$.
 * The image of the homomorphism from $$N$$ to $$G$$ equals the kernel of the homomorphism from $$G$$ to $$Q$$.

Relationship with group extensions
We can think of short exact sequences as being informationally equivalent to group extensions. Explicitly, given a short exact sequence of the form:

$$1 \to N \to G \to Q \to 1$$

we can think of $$G$$ as a group extension with "normal subgroup" $$N$$ and "quotient group" $$Q$$.