Short exact sequence of groups

Definition

A short exact sequence of groups is an 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" $Q$.