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{{for|the finite simple group|Higman–Sims group}} ...]] with no [[trivial group|nontrivial]] [[finite group|finite]] [[quotient group|quotients]]. ...

...S \to K(S)</math>. The space <math>K(S)</math> may be described by the [[K-theory spectrum]] of ''S''. equivalence of all infinite loop space machines<ref>[https://www.ams.org/notices/199608/comm-thomason.p ...

In the [[mathematics|mathematical]] field of [[group theory]], a group is '''residually ''X''''' (where ''X'' is some property of groups) if it "c ...''g'' there is a [[Group homomorphism|homomorphism]] ''h'' from ''G'' to a group with property ''X'' such that <math>h(g)\neq e</math>. ...

...is a [[subgroup]] that determines much of the structure of its containing group. The concept was generalized to [[essential submodule]]s. ...>S</math> of a (typically [[abelian group|abelian]]) [[Group (mathematics)|group]] <math>G</math> is said to be '''essential''' if whenever ''H'' is a non-t ...

..., is an algebra similar to the [[group ring|group algebra]] of a [[Coxeter group]] except that the generators are [[nilpotent]]. The nil-Coxeter algebra for the infinite [[symmetric group]] is the algebra generated by ''u''₁,&nbsp;''u''₂,&nb ...

{{Short description|Infinite Abelian group}} ...f an infinite [[Abelian group]] which is a building block in the structure theory of such groups. ...

{{Short description|A subgroup of a group}} ...substituting group elements for variables in a given set of [[word (group theory)|words]]. ...

...al]] field of infinite [[group theory]], the '''Nottingham group''' is the group ''J''('''F'''_''p'') or ''N''('''F'''_''p'') consistin with coefficients in '''F'''_''p''. The group multiplication is given by formal composition also called substitution. Tha ...

...entation|representation]] <math>(\pi, V)</math> of a [[group (mathematics)|group]] ''G'' is a representation <math>(\pi|_W, W)</math> such that ''W'' is a [ ...tical induction|induction]] on dimension. This fact is generally false for infinite-dimensional representations. ...

...Lifts an action of a finite-dimensional Lie algebra on a manifold to a Lie group action}} ...eorem''' is a partial converse to the fact that any smooth action of a Lie group induces an infinitesimal action of its Lie algebra. {{harvs|txt|last=Palais ...

In mathematics, and in particular [[singularity theory]], an '''{{mvar|A{{sub|k}}}} singularity''', where {{math|''k'' ≥ 0}} is an ...rphisms and <math>f: \R^n \to \R</math> any smooth function. We define the group action as follows: ...

...s, for a [[natural number]] <math>n \ge 2</math>, the ''n''th '''Fibonacci group''', denoted <math>F(2,n)</math> or sometimes <math>F(n)</math>, is defined These [[group (mathematics)|groups]] were introduced by [[John Conway]] in 1965. ...

...ty''' is any of a number of theorems asserting that the [[group cohomology|group homology]] of a series of groups <math>G_1 \subset G_2 \subset \cdots </mat ...ps ''K''_''i'' of rings of algebraic integers.|title=Algebraic K-theory, I: Higher K-theories|pages=179–198|series=Lecture Notes in Math.|volume=34 ...

...geometrical structure (a [[simplicial complex]]) associated to a [[Coxeter group]]. Coxeter complexes are the basic objects that allow the construction of [ Let <math>(W,S)</math> be a [[Coxeter group|Coxeter system]] with [[Coxeter group#Coxeter matrix and Schläfli matrix|Coxeter matrix]] <math> M = (m(s,t))_{s, ...

...bility measure to a larger [[σ-algebra]]. It is of particular interest for infinite dimensional spaces. ...athcal{S})</math>. For instance when <math>\mathcal{S}</math> is countably infinite, this is not always possible. Bierlein's extension theorem says, that it is ...

...also in '''Ulm's theorem''', which describes the classification of certain infinite abelian groups in terms of their '''Ulm factors''' or '''Ulm invariants'''. Let ''A'' be an abelian group and ''g'' an element of ''A''. The ''' ''p''-height''' of ''g'' in ''A'', d ...

In [[mathematics]], in the field of [[algebraic number theory]], a '''modulus''' (plural '''moduli''') (or '''cycle''',<ref>{{harvnb|Lang ...lex places. If ''K'' is a function field, ν('''p''')&nbsp;=&nbsp;0 for all infinite places. ...

{{Short description|special kind of topological group - translation from the German Wikipedia}} ...logical group]] that can be written in a certain sense as a limit of [[Lie group]]s.<ref> Hofmann, Morris 2007, p. vii </ref> ...

...t description|The probability that two uniform random elements of a finite group commute with each other}} ...an group|abelian]] a finite group is. It can be generalized to infinite [[group (mathematics)|groups]] equipped with a suitable [[probability measure]],<re ...

==Group operation== ...th ''x''&nbsp;+&nbsp;''iy'' and ''t''&nbsp;+&nbsp;''iu'' respectively, the group product above is just the ordinary complex number multiplication (''x''&nbs ...