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...thor2=Yakimov, M. T.|title=Unipotent and Nakayama automorphisms of quantum nilpotent algebras|date=1 Nov 2013|class=math.QA|eprint=1311.0278}}</ref> a concept r ...A. Albert|title=Structure of Algebras|page=22|chapter=Chapt. 2: Ideals and Nilpotent Algebras|orig-year=1939|year=2003|series=Colloquium Publications, Col. 24|p ...

...'''' (where ''X'' is some property of groups) if it "can be recovered from groups with property ''X''". * Residually [[nilpotent group|nilpotent]] ...

In [[algebra]], the '''nilradical''' of a [[Lie algebra]] is a nilpotent [[Ideal (Lie algebra)|ideal]], which is as large as possible. ...otent ideal]], which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the [[radical of a Lie algebra|radical]] <math>\mathfrak ...

In [[topology]], a branch of [[mathematics]], a '''nilpotent space''', first defined by Emmanuel Dror Farjoun (1969),<ref>{{cite book|ti * the [[fundamental group]] <math>\pi = \pi_1 (X)</math> is a [[nilpotent group]]; ...

...for some [[prime number]] ''p''. An elementary group is [[nilpotent group|nilpotent]]. * J.P. Serre, "Linear representations of finite groups". Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York, Heidel ...

...scription|On when an element of the coefficient ring of a ring spectrum is nilpotent}} ...ath display="inline">\pi_\ast R \to \mathrm{MU}_\ast(R)</math> consists of nilpotent elements.<ref>{{Cite web |last=Lurie |first=Jacob |date=April 27, 2010 |tit ...

....}}</ref> Equivalently, a Lie algebra is toral if it contains no nonzero [[nilpotent]] elements. Over an algebraically closed field, every toral Lie algebra is ...k g</math> has only nilpotent elements, then it is [[nilpotent Lie algebra|nilpotent]] ([[Engel's theorem]]), but then its [[Killing form]] is identically zero, ...

...Ultralimit]]) of finitely-generated nilpotent groups, and of nilpotent Lie groups, as well as tangent cones of [[sub-Riemannian manifold]]s. Carnot groups were introduced, under that name, by {{harvs|txt|last=Pansu|first=Pierre|au ...

{{Short description|Collection of groups}} ...oup|commutativity]]). Since [[set theory]] does not admit the "set of all groups", it is necessary to work with the more general concept of ''[[Class_(set_t ...

...hematics, the '''Jacobson&ndash;Morozov theorem''' is the assertion that [[nilpotent]] elements in a semi-simple [[Lie algebra]] can be extended to [[sl2-triple ...nilpotent. The Jacobson&ndash;Morozov theorem states that, conversely, any nilpotent non-zero element <math>e \in \mathfrak g</math> can be extended to an sl<su ...

...', is a generalization of a rational nilpotent [[Lie algebra]], and Malcev groups are similar. Both were introduced by {{harvtxt|Quillen|1969|loc=Appendix A3 ...|Quillen|1969|loc=Appendix A3}} showed that Malcev Lie algebras and Malcev groups are both equivalent to complete [[Hopf algebra]]s, i.e., Hopf algebras ''H' ...

...p theory]], a branch of [[mathematics]], a '''formation''' is a [[class of groups]] closed under taking images and such that if ''G''/''M'' and ''G''/''N'' a ...', the formation of π-groups for a set of primes π, and the formation of [[nilpotent group]]s. ...

...]], to study the group. It is mostly of interest for the study of infinite groups. ...p|finitely generated]] and [[finitely presented group|finitely presented]] groups. ...

Examples of approximate subgroups which are not groups are given by symmetric intervals and more generally [[arithmetic progressio ...l example is given by balls in the [[word metric]] in finitely generated [[nilpotent group]]s. ...

....<ref>{{cite book|author=Müller, Detlef|chapter=Functional calculus on Lie groups and wave propagation|title=Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 19 ...179590-7|title=A homogeneous, globally solvable differential operator on a nilpotent Lie group which has no tempered fundamental solution|journal=Proceedings of ...

...<math> \phi </math> that fits into the [[short exact sequence]] of abelian groups: ...eld#Generalization: algebra over a ring|algebra]] over ''A''" and "abelian groups" with "''A''-[[Module (mathematics)|modules]]". ...

...=Philip | author1-link=Philip Hall | title=The Edmonton notes on nilpotent groups. Notes of lectures given at the Canadian Mathematical Congress Summer Semin ...

* <math>{\mathfrak g}_{3.1}</math>, Heisenberg–Weyl algebra, nilpotent, Bianchi II, * <math>{\mathfrak g}_{3.1} \oplus {\mathfrak g}_1</math>, decomposable nilpotent, ...

| title = Rationally varying polarizing subalgebras in nilpotent Lie algebras }}</ref> as well as in [[harmonic analysis]] on Lie groups and [[mathematical physics]]. ...

.../''J''(''R'') is [[semisimple algebra|semisimple]] and ''J''(''R'') is a [[nilpotent ideal]], where ''J''(''R'') denotes the [[Jacobson radical]]. The [[theore ...because <math>R/J</math> is a semisimple ring. Furthermore, since ''J'' is nilpotent, only finitely many of the <math>F_i</math> are nonzero. If ''M'' is Artini ...