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The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits (which implies it right-splits, and corresponds to a direct product). This article takes the latter ...
For a left split sequence, the map t × r: B → A × C gives an isomorphism, so B is a direct sum (3.), and thus inverting the isomorphism and composing with the natural injection C → A × C gives an injection C → B splitting r (2.). However, if a short exact sequence of groups is right split (2.), then it need not be left split or a ...
If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (that is, an exact sequence indexed by the natural numbers) on homology by application of the zig ...
The question of what groups are extensions of by is called the extension problem, and has been studied heavily since the late nineteenth century.As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups {}, where each {+} is an extension of {} by some simple group.
A version of the splitting lemma for groups states that a group G is isomorphic to a semidirect product of the two groups N and H if and only if there exists a short exact sequence 1 N β G α H 1 {\displaystyle 1\longrightarrow N\,{\overset {\beta }{\longrightarrow }}\,G\,{\overset {\alpha }{\longrightarrow }}\,H\longrightarrow 1}
Splitting, also called binary thinking, dichotomous thinking, black-and-white thinking, all-or-nothing thinking, or thinking in extremes, is the failure in a person's thinking to bring together the dichotomy of both perceived positive and negative qualities of something into a cohesive, realistic whole.
From this we see that F is an exact functor if and only if R 1 F = 0; so in a sense the right derived functors of F measure "how far" F is from being exact. If the object A in the above short exact sequence is injective, then the sequence splits. Applying any additive functor to a split sequence results in a split sequence, so in particular R 1 ...
The preceding interpretation of outer automorphisms as symmetries of a Dynkin diagram follows from the general fact, that for a complex or real simple Lie algebra, 𝔤, the automorphism group Aut(𝔤) is a semidirect product of Inn(𝔤) and Out(𝔤); i.e., the short exact sequence. 1 Inn(𝔤) Aut(𝔤) Out(𝔤) 1. splits.