Exact sequence of groups

Definition

Over the integers

An exact sequence of groups is defined as the following data:

• A collection $G_{n}$ of groups, where $n$ varies over the integers
• A collection $d_{n}:G_{n}\to G_{n-1}$ of group homomorphisms

such that for any $n$, the kernel of $d_{n-1}$ equals the image of $d_{n}$.

The exact sequence is typically written as:

$\dots \to G_{n}{\stackrel {d_{n}}{\to }}G_{n-1}{\stackrel {d_{n-1}}{\to }}G_{n-2}\to \dots$

Note that an exact sequence is a particular kind of chain complex of groups.