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...nce of [[Philip Hall]] and [[Ernst Witt]]'s [[Commutator#Identities (group theory)|eponymous identity]]. In what follows, the following notation will be employed: ...

{{Short description|On a particular long exact sequence in the homology groups of certain chain complexes}} ...e [[homology group]]s of certain [[chain complex]]es. The result is valid in every [[abelian category]]. ...

...] had discovered it earlier, in 1953.<ref>{{citation|series=Lectures Notes in Mathematics|volume=1758|year=2001|title=Continuous Bounded Cohomology of Lo ==Statement for group rings== ...

...ds. Hall trees can also be used to give a total order to the elements of a group, via the [[commutator collecting process]], which is a special case of the ...ng process]] was described first, in 1934, by [[Philip Hall]] and explored in 1937 by [[Wilhelm Magnus]].<ref> ...

...in ergodic theory, for example in [[Ornstein isomorphism theorem|Ornstein theory]] and has many generalizations. ...is which are traditionally called lemmas despite the fact that their roles in their respective fields are fundamental. ...

{{Short description|Theorem in abstract algebra}} ...newspaper=[[Time (magazine)|Time]] |access-date=December 14, 2009 }}</ref> In 2010, Ngô was awarded the [[Fields Medal]] for this proof. ...

...ubjected to certain [[finite set|finiteness]] conditions on [[chain (order theory)|chain]]s of [[subgroup]]s, can be uniquely written as a finite [[direct pr We say that a group ''G'' satisfies the [[ascending chain condition]] (ACC) on subgroups if eve ...

{{Short description|Method in combinatorics}} ...], and [[Ramsey theory]]; they include some of the most classical problems in the associated fields. ...

...ting set of a group|generates]] a [[free group|free]] [[subgroup]] of that group. ...variations are widely used in [[geometric topology]] and [[geometric group theory]]. ...

...al proof|proof]] of the [[Hahn embedding theorem]] and then studied by him in relation to [[Hilbert's second problem]]. ...field <math>K</math> and with value group <math>\Gamma</math> (an ordered group) is the set of formal expressions of the form ...

{{About|open mapping theorem in [[functional analysis]]|other results with the same name|Open mapping theor In [[functional analysis]], the '''open mapping theorem''', also known as the ...

...ion is from an '''exceptional collection''', a special sequence of objects in a triangulated category. For an [[algebraic variety]] ''X'', it has been fr ...eq n</math> and all objects <math>A_i\in\mathcal{A}_i</math> and <math>A_j\in\mathcal{A}_j</math>, every morphism from <math>A_j</math> to <math>A_i</mat ...

In [[abstract algebra]], '''Kaplansky's theorem on projective modules''', firs ..."is very likely the inspiration for a major portion of the results" in the theory of [[semiperfect ring]]s.<ref name="theorem" /> ...

In the [[mathematics|mathematical]] area of [[geometric group theory]], a '''Van Kampen diagram''' (sometimes also called a '''Lyndon–Van Kampen ...en by a [[group presentation]] represents the [[identity element]] in that group. ...

In [[abstract algebra]], a '''decomposition of a module''' is a way to write a ...re equivalent to each other; a special case of this, especially in [[group theory]], is known as the [[Krull–Schmidt theorem]]. ...

...d by Gessel–Viennot in 1985, based on previous work of Lindström published in 1973. The lemma is named after [[Bernt Lindström]], [[Ira Gessel]] and [[Gé ...ned if between any two points there are only finitely many paths; but even in the general case, this can be well-defined under some circumstances (such a ...

{{Short description|Theorem in graph theory}} ...s. It is a generalization of the [[graph removal lemma]]. The special case in which the graph is a tetrahedron is known as the [[tetrahedron removal lemm ...

...an output signal based on a threshold function. By connecting these units in various configurations, McCulloch and Pitts demonstrated that their model c ...ical theory of automata]''. In L. A. Jeffress (Ed.), ''Cerebral mechanisms in behavior; the Hixon Symposium'' (pp. 1–41). Wiley.</ref> ...

This is a glossary of properties and concepts in [[algebraic topology]] in mathematics. ...topology]], [[list of algebraic topology topics]], [[glossary of category theory]], [[glossary of differential geometry and topology]], [[Timeline of manifo ...

...ey are predicted to exist through [[mathematical model]]s and [[scientific theory]].]] ...lain, and predict [[List of natural phenomena|natural phenomena]]. This is in contrast to [[experimental physics]], which uses experimental tools to prob ...