nonsplit

Nonsplit is an adjective used primarily in mathematics to describe a structure, extension, or object that cannot be decomposed into a direct product, direct sum, or semidirect product in a way that respects the given structure.

In the context of exact sequences and extensions, a short exact sequence 0 → A → B → C → 0 is said to split if there exists a homomorphism s: C → B with the property that the composition of the projection B → C with s is the identity on C. When such a splitting homomorphism exists, B is isomorphic to the direct sum (or direct product) A ⊕ C (in abelian settings) or to a semidirect product in groups. If no such splitting exists, the sequence is nonsplit. A classical example is the sequence 0 → Z/2Z → Z/4Z → Z/2Z → 0, which is nonsplit because Z/4Z is not isomorphic to Z/2Z ⊕ Z/2Z.

In module theory and group theory, nonsplit extensions indicate that the interaction between the subobject and quotient is nontrivial and cannot be separated into independent components. The term is also used in the study of algebraic groups over a field: a group form can be called nonsplit if it does not contain a split maximal torus over the field, indicating a more intricate internal structure than a split form.

Nonsplit is a descriptive label rather than a standalone object, and its precise meaning depends on the mathematical context—often relating to the absence of a splitting map or decomposition.