Exact sequence of groups

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.