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In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective. More formally, proposition B is a corollary of proposition A, if B can be readily deduced from A or is self-evident from its proof.
Whether the snake lemma holds in the category of groups depends on the definition of cokernel. If f : A → B {\displaystyle f:A\to B} is a homomorphism of groups, the universal property of the cokernel is satisfied by the natural map B → B / N ( im f ) {\displaystyle B\to B/N(\operatorname {im} f)} , where N ( im f ) {\displaystyle N ...
The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears. [ 4 ] Statement 3 : If M {\displaystyle M} is a finitely generated module over R {\displaystyle R} , J ( R ) {\displaystyle J(R)} is the Jacobson radical of R {\displaystyle R} , and J ( R ) M = M {\displaystyle J(R)M=M} , then M = 0 ...
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem".
As a corollary, if the index of H in G is 2, or for a finite group the lowest prime p that divides the order of G, then H is normal, as the index of its core must also be p, and thus H equals its core, i.e., it is normal.
An edge e is free if and only if, in an arbitrary maximum matching M, edge e belongs to an even alternating path starting at an unmatched vertex or to an alternating cycle. By the first corollary, if edge e is part of such an alternating chain, then a new maximum matching, M ′, must exist and e would exist either in M or M ′, and
In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, ... Corollary [1] — If : is smooth ...
In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. For example, if there is a party of people who shake hands, the number of people who shake an odd number of other people's hands is even. [ 1 ]