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...=Philip | author1-link=Philip Hall | title=A contribution to the theory of groups of prime-power order | year=1934 | journal=Proceedings of the London Mathem The process can be generalized to define a [[totally ordered]] subset of a free non-associative algebra, that is, a [[free magma]]; this ...

{{Short description|Equivalence of partially ordered sets}} ...on]] that constitutes a suitable notion of [[isomorphism]] for [[partially ordered set]]s (posets). Whenever two posets are order isomorphic, they can be cons ...

* The [[Partially ordered set|partial order]] relations of divisibility are [[Lattice (order)|lattice ...s | title=Gaussian groups and Garside groups, two generalisations of Artin groups | journal=[[Proceedings of the London Mathematical Society]] | volume=79 | ...

...d]]s, varieties of finite [[ordered semigroup]]s and varieties of finite [[ordered monoid]]s are defined similarly. A variety ''V'' of finite (ordered) semigroups is a class of finite (ordered) semigroups that: ...

...h>S</math> if it [[group action|acts]] transitively on the set of distinct ordered pairs <math>\{ (x,y) \in S \times S : x \neq y \}</math>. That is, assuming ...'i''</sub> for each ''i'' between 1 and ''k''. The [[Mathieu group|Mathieu groups]] are important examples. ...

...finite group is. It can be generalized to infinite [[group (mathematics)|groups]] equipped with a suitable [[probability measure]],<ref name=":0" /> and ca ...t}}</ref>) and this upper bound is sharp: there are infinitely many finite groups <math>G</math> such that <math>p(G) = 5/8</math>, the smallest one being th ...

...ee (set theory)|trees]], abelian [[linearly ordered group|linearly ordered groups]], algebraically closed [[valued field|valued fields]], and the [[p-adic nu ...thor2=Pillay, Anand |author3=Steinhorn, Charles | title=Definable sets in ordered structures II | journal=[[Transactions of the American Mathematical Society ...

...lex is called a Poincaré complex if its [[homology group]]s and cohomology groups have the abstract properties of Poincaré duality.<ref name="YUB">{{Springer ...be a [[chain complex]] of [[abelian group]]s, and assume that the homology groups of <math>C</math> are [[Generating set of a group|finitely generated]]. Ass ...

...bras|varieties of algebras]], such as the variety of [[group (mathematics)|groups]], consist of congruence-permutable algebras, but some, like the variety of {{ordered list ...

...nite]]; this explains the terminology. Basic examples of locally profinite groups are [[discrete group]]s and the [[p-adic Lie group|''p''-adic Lie group]]s. ...ps come from [[algebraic number theory]]. Let ''F'' be a [[Non-Archimedean ordered field|non-archimedean]] [[local field]]. Then both ''F'' and <math>F^\times ...

...or the conjugacy classes of involutions, and ''a''(''x'') is the number of ordered pairs of involutions ''u'',''v'' such that ''u'' is conjugate to ''t'', ''v ...=R. | author1-link=Richard Brauer | last2=Fowler | first2=K. A. | title=On groups of even order | jstor=1970080 |mr=0074414 | year=1955 | journal=[[Annals of ...

...sses that <math>a \leq c</math></ref> <math>M</math> is called ''naturally ordered'' if the <math>\leq</math> relation is additionally [[antisymmetric relatio ...not a partial order. There are also examples of monoids that are naturally ordered but are not semirings with monus.<ref>{{cite web ...

...n the way. In certain situations (such as descent theorems for fundamental groups à la Van Kampen) it is much more elegant, even indispensable for understand ...upoid {{math|Π(''X'')}}, or {{math|Π₁(''X'')}}, assigns to each ordered pair of points {{math|(''p'', ''q'')}} in {{mvar|X}} the collection of equi ...

...its opposite. {{angbr|''g''₁, ''g''₂}} denotes the [[ordered pair]] of the two group elements. *' can be viewed as the naturally induced [[Monoid|Monoids]], groups, [[Ring (mathematics)|rings]], and [[Algebra over a ring|algebras]] can be ...

...an have only one identity and any element can have only one inverse, right groups allow for multiple {{nowrap|one-sided}} [[identity element]]s and multiple ...ill be mostly concerned about right groups, but everything applies to left groups by doing the appropriate right/left substitutions. ...

* {{annotated link|Ordered topological vector space}} [[Category:Topological groups]] ...

...R_{>0}, >\right)</math> forms a [[total order]] but is {{em|not}} a [[well-ordered set]]. The [[doubly infinite]] [[geometric progression]] <math>10^n,</math> ...ebesgue measure is invariant under addition. In the context of topological groups, this measure is an example of a [[Haar measure]]. ...

===Partially ordered sets and lattices=== [[Ordered vector space|Partially ordered vector space]]s and [[Riesz space|vector lattices]] are used throughout sta ...

{{ordered list | list-style-type = lower-roman ...ng the [[stack quotient]] of some variety where the stabilizers are finite groups. For example, consider the action of the cyclic group <math>C_n = \langle a ...

...epresentation has several foundational properties in the theory of Coxeter groups; for instance, <math>B</math> is positive definite if and only if <math>W</ ===Finite dihedral groups=== ...