A short exact sequence of groups
(1)
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is called split if it essentially presents as the direct sum of the groups and .
More precisely, one can construct a commutative diagram as diagrammed above, where is the injection of the first summand and is the projection onto the second summand , and the vertical maps are isomorphisms.
Not all short exact sequences of groups are split. For example the short exact sequence diagrammed above cannot be split, since and are non isomorphic finite groups. Note that this is also a short exact sequence of -modules: this shows that being split is a distinguished property of short exact sequences also in the category of modules. In fact, it is related to particular classes of modules.
Given a module over a unit ring , all short exact sequences
(2)
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are split iff is projective, and all short exact sequences
(3)
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A short exact sequence of vector spaces is always split.