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...'''' (where ''X'' is some property of groups) if it "can be recovered from groups with property ''X''". ...if it embeds into its pro-''X'' completion (see [[profinite group]], [[pro-p group]]), that is, the [[inverse limit]] of the [[inverse system]] consisti ...

...igman–Sims asymptotic formula''' gives an asymptotic estimate on number of groups of prime power order. ...(n,p)</math> as the number of isomorphism classes of groups of order <math>p^n</math>. Then: ...

...cteristic 2 type, where involutions resemble unipotent elements, and other groups, where involutions resemble semisimple elements. Groups of characteristic 2 type and rank at least 3 are classified by the [[tricho ...

{{Short description|Manifold with the same rational homotopy groups as a sphere}} ...ing [[Torsion (algebra)|torsion]] in comparison to the (integral) homotopy groups of the space. ...

...ster}} He called them "perfect groups"{{r|leinster}} and later "immaculate groups",{{r|brunault}} but they were renamed as the Leinster groups by {{harvtxt|De Medts|Maróti|2013}} because "[[perfect group]]" already had ...

{{Short description|Manifold with the same rational homology groups as a sphere}} ...ing [[Torsion (algebra)|torsion]] in comparison to the (integral) homology groups of the space. ...

...are used in the theory of [[Lattice (discrete subgroup)|lattices]] in Lie groups, often under the name ''field of definition''. == Fuchsian and Kleinian groups == ...

...s | title=Gaussian groups and Garside groups, two generalisations of Artin groups | journal=[[Proceedings of the London Mathematical Society]] | volume=79 | ...citation | last=Garside | first=Frank A. | title=The braid group and other groups | journal=Quarterly Journal of Mathematics |series=Second Series | volume=2 ...

...ization of distributions''' says that every [[probability distribution]] ''P'' admits (in the convolution semi-group of probability distributions) a fac :<math>P = P_1 \otimes P_2 </math> ...

{{about|graphs defined from finite groups|other uses|Glossary of graph theory#prime}} ...n [[edge (graph theory)|edge]] connecting each pair of prime numbers <math>p</math> and <math>q</math> for which there exists a group element with order ...

:ATP + X + Y <math>\rightleftharpoons</math> AMP + X-''P'' + Y-''P''. ...rases with paired acceptors (dikinases)|2.7.9]]. It is one of the smallest groups of phosphotransferases. ...

...an elementary group if it is ''p''-elementary for some [[prime number]] ''p''. An elementary group is [[nilpotent group|nilpotent]]. More generally, a finite group ''G'' is called a ''p''-'''hyperelementary''' if it has the [[group extension|extension]] ...

...her the centroid and/or the spread of the objects is different between the groups. Hence the test is based on the prior calculation of the distance between a ...result to that gained from random permutations of the objects between the groups. Moreover, whilst PERMANOVA tests for similarity based on a chosen distance ...

...t1=Brauer |first1=Richard D. |year=1963 |chapter=Representations of finite groups |title=Lectures in Mathematics |volume=1 |publisher=Wiley | pages=133–175 | ...bgroups]], called the [[Modular representation theory#Defect groups|defect groups]] of <math>B</math>. The set of irreducible characters belonging to <math>B ...

...cyclic]] of order dividing <math>p^f-1</math>, and <math>R</math> of <math>p</math>-power order. Therefore, <math>\text{Gal}(L/\mathbf{Q}_p)</math> is s [[Category:Properties of groups]] ...

...ting process]], and implies that ''p''-groups of small class are [[regular p-group|regular]]. ...arshall | author1-link=Marshall Hall (mathematician) | title=The theory of groups | publisher=Macmillan |mr=0103215 | year=1959}} ...

...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 ...roup|''p''-groups]] for a [[prime number|prime]] ''p'', the formation of π-groups for a set of primes π, and the formation of [[nilpotent group]]s. ...

...alance, but the proof of this requires the classification of finite simple groups (more precisely the [[Schreier conjecture]]). ...n | last2=Walter | first2=John H. | title=Balance and generation in finite groups | doi=10.1016/0021-8693(75)90123-4 | mr=0357583 | year=1975 | journal=[[Jou ...

...]]. A simplicial group is a [[Kan complex]] (in particular, its [[homotopy groups]] make sense). The [[Dold–Kan correspondence]] says that a simplicial abeli A [[commutative monoid]] in the category of simplicial abelian groups is a [[simplicial commutative ring]]. ...

...n [[Daniel Quillen|Quillen]]'s work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic leve ...it]] of the chromatic tower (defined below) of a finite [[local spectrum|''p''-local spectrum]] <math>X</math> is <math>X</math> itself. The theorem was ...