Exact sequence of groups

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Definition

Over the integers

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

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.