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For a direct sum this is clear, as one can inject from or project to the summands. 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.).
A split extension is an extension 1 → K → G → H → 1 {\displaystyle 1\to K\to G\to H\to 1} with a homomorphism s : H → G {\displaystyle s\colon H\to G} such that going from H to G by s and then back to H by the quotient map of the short exact sequence induces the identity map on H i.e., π ∘ s = i d H {\displaystyle \pi \circ s ...
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 ...
An extension L that is a splitting field for a set of polynomials p(X) over K is called a normal extension of K.. Given an algebraically closed field A containing K, there is a unique splitting field L of p between K and A, generated by the roots of p.
Pages in category "Abstract algebra" The following 143 pages are in this category, out of 143 total. ... Split exact sequence; Subfield of an algebra; Subquotient; T.
In other words, there is a split exact sequence of groups (which is also known as a split extension of by ). If any of these statements holds (and hence all of them hold, by their equivalence), we say G is the semidirect product of N and H , written