A short exact sequence of groups A, B, and C is given by two maps α:A->B and β:B->C and is written 0->A->B->C->0. Because it is an exact sequence, α is injective, and β is surjective. Moreover, the group kernel of β is the image of α. Hence, the group A can be considered as a (normal) subgroup of B, and C is isomorphic to B/A. A short exact sequence is said to split if there is a map γ:C->B such that β°γ is the identity on C. This only happens when B is the direct product of A and C. The notion of a short exact sequence also makes sense for modules and sheaves.

exact sequence | group extension | long exact sequence | module | principal bundle | split exact sequence