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...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 ...

{{for|the Baer norm (intersection of normalizers)|Norm (group)}} ...math>(G, +)</math> is an ([[abelian group|abelian]]) [[Group_(mathematics)|group]] with identity element <math>e</math> then <math>\nu\colon G \to \mathbb{R ...

...ometry, an '''abelian Lie group''' is a [[Lie group]] that is an [[abelian group]]. ...p]] that is a compact group is abelian and a connected compact complex Lie group is a [[complex torus]]; i.e., a quotient of <math>\mathbb{\Complex}^n</math ...

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

...quirement that every morphism have a kernel and cokernel, as is true for [[abelian categories]]. Synonyms in the literature for pseudo-abelian include '''pseudoabelian''' and '''Karoubian'''. ...

...thful action|faithful]] since <math>S</math> has a trivial [[Center (group theory)|center]].<ref>{{Cite journal |last1=Dallavolta |first1=F. |last2=Lucchini ...>\mathrm{S}_n</math> is the automorphism group of the simple [[alternating group]] <math>\mathrm{A}_n,</math> so <math>\mathrm{S}_n</math> is almost simple ...

...entary for some [[prime number]] ''p''. An elementary group is [[nilpotent group|nilpotent]]. ...auer's theorem on induced characters]] states that a character on a finite group is a linear combination with integer coefficients of characters [[induced c ...

...ake sense). The [[Dold–Kan correspondence]] says that a simplicial abelian group may be identified with a [[chain complex]].<!-- need more precision --> In any simplicial abelian group <math>A</math> is non-canonically [[homotopy equivalent]] to a product of [ ...

...'₁, ''g''₂}} denotes the [[ordered pair]] of the two group elements. *' can be viewed as the naturally induced addition of +.]] ...define [[Group action (mathematics)|right action]] as a special case of [[Group action (mathematics)|left action]]. ...

In [[group theory]], a '''balanced group''' is a [[topological group]] whose left and right [[uniform space|uniform structres]] coincide. A topological group <math>G</math> is said to be '''balanced''' if it satisfies the following e ...

{{Short description|Maximal abelian extension of an algebraic number field}} ...' is the degree [''Γ(K)'':''K''] and the '''genus group''' is the [[Galois group]] of ''Γ(K)'' over ''K''. ...

...s <math>\Gamma(U, \mathcal{F})</math> is a [[torsion group|torsion abelian group]]. Similarly, for a [[prime number]] ''p'', we say a sheaf <math>\mathcal{F *{{cite book |first=Lei|last=Fu|doi=10.1142/9569|title=Etale Cohomology Theory |series=Nankai Tracts in Mathematics |year=2015 |volume=14 |isbn=978-981-46 ...

'''Gaussian distribution on a locally compact Abelian group''' is a [[Probability distribution|distribution]] <math>\gamma</math> on a countable [[Locally compact abelian group|locally compact Abelian group]] <math>X</math> which satisfies the ...

{{Short description|An abelian group}} ...le group]]. This abelian group is a [[Klein four-group]]-module, where the group acts by reflection in each of the coordinate directions (here depicted by r ...

...mathematics)|ramification]] in the field generated by the [[Division point#Abelian varieties|torsion points]]. For an [[abelian variety]] ''A'' defined over a field ''F'' as above, with ring of integers ...

...description|Simplicial commutative ring: commutative monoid in simplicial abelian groups}} {{mergeto|Simplicial group|date=July 2024}} ...

{{Short description|Theorem in probability theory}} ...theory]]. It is well known that if each of two [[independence (probability theory)|independent]] [[random variables]] ξ₁ and ξ₂ has a [ ...

{{short description|Concept in group theory}} ...ract algebra]] known as [[group theory]], the '''diameter''' of a [[finite group]] is a measure of its complexity. ...

{{Short description|Group theoretic operation}} ...Rosati]], is an involution of the rational [[endomorphism ring]] of an [[abelian variety]] induced by a [[Abelian_variety#Polarisations|polarisation]]. ...

...''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 ...