The same as an inverse semi-group (cf. [[Inversion semi-group|Inversion semi-group]]).
[[Category:Group theory and generalizations]]
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''dihedron group''
...In a finite group, two different elements of order 2 generate a dihedral group.
1 KB (202 words) - 16:24, 19 October 2014
...sentation of $G$ on the set of right cosets of $H$ in $G$ (cf [[Coset in a group]]). Then its kernel is the core of $H$ in $G$.
.../TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.R. Scott, "Group theory" , Dover, reprint (1987) (Original: Prentice-Hall, 1964)</TD></TR></table
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...[Group|Group]]), where the [[Frattini-subgroup(2)|Frattini subgroup]] of a group plays an important role. Due to the many close connections which Lie algebr
...and algebras]]) is the intersection of its maximal subalgebras. Unlike the group case, where the Frattini subgroup is always a [[Normal subgroup|normal subg
2 KB (323 words) - 13:52, 25 April 2014
''equi-affine group''
The subgroup of the general [[affine group]] consisting of the affine transformations of the $n$-dimensional affine sp
1 KB (158 words) - 22:38, 2 November 2014
...between the elements of any finite generating set of a finitely-presented
group contains a finite set of defining relations in these generators.
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The
theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian
2 KB (242 words) - 18:26, 26 October 2014
...perators on $E$. The space $E$ is called the representation space of $\pi$ and the operators $\pi(x)$, $x\in X$, are called the operators of the represent
...valign="top">[1]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)</TD></TR>
1 KB (197 words) - 19:18, 25 October 2014
...gebras was discovered by C. Chevalley [[#References|[2]]] (cf. [[Chevalley group]]). In particular, Chevalley's method makes it possible to obtain Dickson g
[[Category:Group theory and generalizations]]
1 KB (187 words) - 21:05, 15 November 2017
...of $G$ coincides with the rank of $\widehat G$ (cf. [[Rank of an algebraic group]]).
...">[a2]</TD> <TD valign="top"> V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) {{MR|0376938}} {{ZBL|0371.220
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...mi-group]] with unit (i.e. [[Monoid]]) satisfying the [[cancellation law]] and in which any non-invertible element $a$ is decomposable into a product of i
a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l
1 KB (153 words) - 16:17, 21 December 2014
...). As an operation on a class of groups, the verbal product is associative and, within the corresponding [[variety of groups]], it is also free.
...litar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966) {{ZBL|0138.25604}}. pp. 412</TD><
883 bytes (142 words) - 19:33, 11 December 2015
...r groups are said to be cyclic (they are isomorphic to either the additive group $\mathbf Z$ of integers, or the additive groups $\mathbf Z_n$ of residue cl
...groups that are simple (cf. [[Finitely-presented group|Finitely-presented group]]).
2 KB (343 words) - 18:24, 26 October 2014
''Lie group of type $(E)$''
A real finite-dimensional [[Lie group|Lie group]] $G$ for which the [[Exponential mapping|exponential mapping]] $\exp\colon
2 KB (253 words) - 22:20, 14 November 2014
''triangular Lie group''
...f. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]) are real for any element $g$.
2 KB (303 words) - 18:20, 12 December 2019
...as an increasing ideal series (see [[Ideal series|Ideal series]] of a semi-group) such that for any two adjacent terms $A_\alpha,A_{\alpha+1}$,
...for right (or left) ideals. If all nilpotent sub-semi-groups of a nil semi-group $S$ are finite, then so is $S$.
2 KB (360 words) - 17:24, 14 October 2014
''of a group $A$ by a group $B$''
...ruct a semi-direct product one should also know which automorphisms of the group $B$ are induced by conjugation by elements of $A$. More precisely, if $G =
2 KB (359 words) - 16:54, 23 November 2023
''soluble group''
...] of a group). The term "solvable group" arose in [[Galois theory|Galois theory]] in connection with the solvability of algebraic equations by radicals.
3 KB (443 words) - 18:25, 26 October 2014
...of L. Sylow, who proved a number of theorems on such subgroups in a finite group (see [[Sylow theorems|Sylow theorems]]).
...f its Sylow $2$-subgroups. In the theory of infinite groups, except in the theory of locally finite groups, the role of Sylow subgroups is less important, si
2 KB (324 words) - 19:15, 4 April 2023
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...en the derived group of $G$ is locally finite and, in fact, locally normal and Chernikov.
2 KB (350 words) - 16:55, 1 July 2020
...$ with degree prime to $p$ (cf. also [[Character of a group|Character of a group]]). The simplest form of the McKay–Alperin conjectures asserts that
...]]] first suggested this might be true when $G$ is a [[Simple group|simple group]]. J.L. Alperin [[#References|[a1]]] observed that it is probably true for
2 KB (358 words) - 18:45, 13 October 2014
...heory of Lie groups and algebraic groups (cf. [[Lie group, semi-simple|Lie group, semi-simple]]).
...simple. There are examples of simple groups that are not absolutely simple and of simple groups that are not strictly simple.
3 KB (510 words) - 17:22, 14 October 2014
...up, proved by L. Sylow [[#References|[1]]] and playing a major role in the theory of finite groups. Sometimes the union of all three theorems is called Sylow
Let $G$ be a finite group of order $p^ms$, where $p$ is a prime number not dividing $s$. Then the fol
2 KB (372 words) - 19:17, 4 April 2023
...development of the Galilean relativity principle forms part of the general theory of relativity.
.../TD> <TD valign="top"> V.A. [V.A. Fok] Fock, "The theory of space, time and gravitation" , Macmillan (1964) (Translated from Russian)</TD></TR></tabl
2 KB (272 words) - 17:21, 7 February 2011
$#C+1 = 13 : ~/encyclopedia/old_files/data/L058/L.0508600 Lie group, Banach
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3 KB (427 words) - 22:16, 5 June 2020
...tric|Hyperbolic metric]]), $M_3$ is the Euclidean Hantzche–Wendt manifold, and $M_2$ is the [[Lens space|lens space]] $L(5,2)$ (see [[#References|[a2]]]).
...the three-dimensional sphere $S^3$, discovered by H.M. Hilden, M.T. Lozano and J.M. Montesinos [[#References|[a3]]]. In fact,
5 KB (677 words) - 13:19, 26 March 2023
...For any elements $x,y,z\in G$, from $x\leq y$ it follows that $xz\leq yz$ and $zx\leq zy$.
...$ satisfying conditions 1)–4), then $G$ can be made into a totally ordered group with $P$ as set of positive elements.
3 KB (495 words) - 19:18, 17 August 2014
...ons coincide, and in this case one simply speaks of the decomposition of a group with respect to a normal subgroup.
[[Category:Group theory and generalizations]]
1 KB (215 words) - 20:26, 4 December 2014
...|nil semi-group]], or $S$ is a subdirect product of a group and a nil semi-group.
...band of torsion classes; this clearly occurs for commutative semi-groups, and it is true for periodic semi-groups having two idempotents [[#References|[3
5 KB (775 words) - 17:22, 17 October 2014
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...s the presentation (cf. also [[Finitely-presented group|Finitely-presented group]]; [[Presentation|Presentation]]):
6 KB (821 words) - 16:46, 1 July 2020
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...ame ambient isotopy class of oriented links (cf. also [[Braid theory|Braid theory]]), then one can transform one braid to another by a sequence of Markov mov
2 KB (295 words) - 08:04, 19 March 2023
...(E. Noether, W. Krull), whence follow the analogous theorems for invariant and fully-invariant series.
...normal series of elements in a lattice. 4) Generalizations of the Schreier and Jordan–Hölder theorems have been obtained for normal categories (see [[#
4 KB (660 words) - 20:00, 11 April 2014
''group algebra of a group $G$ over a field $K$''
the [[Associative rings and algebras|associative algebra]] over $K$ whose elements are all possible fin
5 KB (872 words) - 22:00, 29 April 2012
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equipped with a transitive transformation group $ G = \{ g \} $
5 KB (684 words) - 08:09, 6 June 2020
...thbf Z$, and $((x+m))=x-1/2$ if $m\in\mathbf Z$, $0<x<1$. For integers $c$ and $d$, with $c>0$, the Dedekind sum $S(d,c)$ is the rational number defined b
...dekind eta-function]] under substitutions from the [[Modular group|modular group]]. This interpretation leads naturally to the reciprocity relation for Dede
3 KB (398 words) - 21:39, 23 December 2015
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...eometry that studies properties of geometrical forms, in particular curves and surfaces, "in the small" . In other words, the structure of a geometrical
4 KB (613 words) - 22:17, 5 June 2020
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of a locally compact group $ G $
13 KB (1,974 words) - 22:12, 5 June 2020
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with a unique non-trivial reduced homology group:
4 KB (616 words) - 19:33, 20 January 2021
...hisms of [[C*-algebra|$C^*$-algebra]]s from the point of view of [[ergodic theory]].
...]) for approximately-finite factors (cf. [[Factor]]) of type $\mathrm{II}$ and type $\mathrm{III}_\lambda$, $0 < \lambda < 1$ (see [[#References|[2]]]).
3 KB (481 words) - 21:35, 15 December 2014
...,\ldots,a_{2m-1}$ is a sequence of elements from the [[Cyclic group|cyclic group]] $\mathbb{Z}_m$, then there exists a set $I\subseteq \{1,\ldots,2m-1\}$ of
zeros and $ m - 1 $
10 KB (1,573 words) - 17:25, 28 January 2020
...dentifying the elements of the block design with the elements of the group and the blocks with the sets $\{d_1g,\dots,d_kg\}$, where $g$ runs over $G$).
...clic difference set a multiplier is a number $t$ relatively prime with $v$ and with the property that
5 KB (832 words) - 10:37, 15 November 2014
...ect to this operation, one can form the [[semi-direct product]] $W$ of $B$ and $A^B$, that is, the set of all pairs $(b,\phi)$, where $b \in B$, $\phi \in
...the wreath product (direct wreath product, discrete wreath product) of $A$ and $B$; it is denoted by $A \mathop{wr} B$ (or $A \wr B$). Both wreath product
10 KB (1,459 words) - 20:29, 27 January 2020
''in inverse Galois theory''
...also [[Galois group|Galois group]]) is a free [[Profinite group|profinite group]] of countable rank. Here, $\textbf{Q}^{\text{ab}}$ is the maximal Abelian
11 KB (1,686 words) - 18:54, 6 February 2021
The divisor class group under algebraic equivalence on a non-singular projective variety.
...ted by $\mathrm{NS}(X)$. The Néron–Severi theorem asserts that the Abelian group $\mathrm{NS}(X)$ is finitely generated.
4 KB (687 words) - 05:47, 15 April 2023
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[[Category:General theory of linear operators]]
4 KB (632 words) - 17:38, 6 January 2024
...es|[2]]]; for an account of the theory of distal dynamic systems and their generalizations, as well as the relevant literature, see [[#References|[3]]].
...tructure theorem) see [[#References|[3]]], (3.15.42), [[#References|[a3]]] and [[#References|[a5]]].
5 KB (686 words) - 11:55, 26 July 2014
...pecially useful for generalizations of the Galois theory of finite, normal and separable field extensions.
...F$ is generated by $P$ and $G$; indeed, it is the [[cross product]] of $P$ and $G$.
3 KB (470 words) - 19:10, 9 November 2016
$#C+1 = 86 : ~/encyclopedia/old_files/data/C023/C.0203180 Cohomotopy group
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7 KB (1,086 words) - 17:45, 4 June 2020
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be a [[Group|group]] of order $ v $
5 KB (803 words) - 06:20, 26 March 2023
The group $\def\Cr#1{{\rm Cr}(\mathbb{P}_k^#1)} \Cr{n}$ of birational automorphisms o
over a field $k$; or, equivalently, the group of Cremona
4 KB (662 words) - 21:51, 30 March 2012
The special linear group
...ant homomorphism $\det_n$. The structure of $\SL(n,R)$ depends on $R$, $n$ and the type of determinant defined on $\GL(n,R)$. There are three main types o
6 KB (988 words) - 22:16, 10 January 2015
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\textrm{ and } \ \mathfrak Z _ {r} ( \mathfrak Z _ {l} ( H) ) = H
7 KB (1,087 words) - 08:09, 6 June 2020
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[[Category:General theory of linear operators]]
5 KB (685 words) - 16:32, 6 January 2024
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Generalizations of the hypergeometric functions $ { {} _ {p} F _ {q} } $
4 KB (590 words) - 11:05, 7 June 2020
...3 : ~/encyclopedia/old_files/data/G110/G.1100050 Gamma\AAhinvariant in the theory of Abelian groups,
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6 KB (845 words) - 19:41, 5 June 2020
...mal subgroups other than $ G $ . A connected Lie group is semi-simple if and only if it splits into a locally direct product of simple non-Abelian norma
...Lie algebras (cf. [[Lie algebra, semi-simple|Lie algebra, semi-simple]]), and also to the global classification of the Lie groups $ G $ that correspon
11 KB (1,458 words) - 18:15, 12 December 2019
...while the latter is based on mappings of arbitrary spaces into polyhedra, and is especially useful whenever the application in fact involves such mapping
...tudy of ordered, rather than of oriented, simplexes by S. Eilenberg (1944) and to cubic homology theories in which cubes, rather than simplices, are emplo
9 KB (1,350 words) - 11:39, 25 October 2014
A basic result in the theory of polynomial identity rings (PI-rings). A ring $R$ is a PI-ring (cf. also
...over a commutative ring satisfies the standard polynomial of degree $2n$, and no polynomial of lower degree.
4 KB (628 words) - 21:20, 11 December 2017
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...discrete topology. Any Hausdorff space can be made into a topological semi-group, e.g. by giving it a left-singular or zero multiplication.
16 KB (2,287 words) - 14:36, 19 March 2023
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A [[Semi-group|semi-group]] in which every element is regular (see [[Regular element|Regular element]
10 KB (1,515 words) - 08:10, 6 June 2020
...(or Euclidean) space; a connection was also established between this fact and the property of an $r$-dimensional manifold (in an $n$-dimensional space) b
...groups by their dual cohomology groups of the same dimension over the dual group of coefficients.
6 KB (855 words) - 06:40, 26 March 2023
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...py equivalence, of certain types of mapping spaces. These results are vast generalizations of two different but related conjectures made by D. Sullivan in 1972.
3 KB (522 words) - 16:57, 1 July 2020
...087670/s08767013.png" /> is a [[balanced incomplete block design]] (BIBD),
and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.o
In 1844 W. Woolhouse stated the existence problem for Steiner systems,
and P. Kirkman solved it in 1847 for <img align="absmiddle" border="0" src="htt
32 KB (4,451 words) - 15:16, 19 March 2018
$#C+1 = 64 : ~/encyclopedia/old_files/data/R080/R.0800520 Reflection group
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12 KB (1,735 words) - 19:59, 15 March 2023
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are two algebraically-irreducible representations of some group or algebra in two vector spaces $ X $
5 KB (720 words) - 08:12, 6 June 2020
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...s of analytic function theory|boundary value problems of analytic function theory]]. It can be stated in the simplest case as follows. Let $ L $
10 KB (1,509 words) - 08:11, 6 June 2020
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...gue of the Riemann hypothesis for elliptic curves by H. Hasse in 1933. The theory of algebraic curves over an arbitrary field of constants, which was develop
8 KB (1,165 words) - 16:08, 1 April 2020
...6 : ~/encyclopedia/old_files/data/G110/G.1100060 Gamma\AAhinvariant in the theory of modular forms,
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7 KB (1,010 words) - 19:41, 5 June 2020
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...unding the complexity of decompositions as graphs of groups for a discrete group.
9 KB (1,405 words) - 15:30, 1 July 2020
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...lowing mathematical concepts (structures) occur in the definitions of Bohr and Bochner almost-periodicity: 1) the space of continuous functions defined on
13 KB (1,809 words) - 19:41, 5 June 2020
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...-theory. In the narrow sense, $ K $-theory is the generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]) g
17 KB (2,459 words) - 07:32, 26 February 2022
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...4]]] on the other. Another approach has been undertaken by Yu. Berezanskii and collaborators [[#References|[a1]]].
9 KB (1,358 words) - 22:11, 5 June 2020
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which is acted upon transitively by the pseudo-unitary group $ \mathop{\rm SU} _ {n,1} $,
5 KB (656 words) - 03:11, 4 January 2022
$#C+1 = 215 : ~/encyclopedia/old_files/data/C022/C.0202370 Class field theory
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17 KB (2,620 words) - 07:48, 13 February 2024
There exist different generalizations of Schauder's theorem: the Markov–Kakutani theorem, Tikhonov's principle,
...">[4]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965) {{MR|0221256}} {{ZBL|
3 KB (485 words) - 20:00, 28 January 2020
...rals are equal, their common value is the Burkill integral of $F$ over $R$ and is denoted by $\int_RF$. If $F$ is integrable over $R$, then $F$ is integra
...ally generalized to set functions with values in a commutative topological group. The Burkill integral is less general than the subsequently introduced [[Ko
3 KB (563 words) - 09:54, 27 November 2018
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that is homogeneous with respect to the group of linear transformations $ \alpha \in \mathop{\rm GL} _ {n} ( \mathbf R
9 KB (1,244 words) - 22:10, 5 June 2020
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...fference set|Difference set]] (Vol. 3). For an extensive discussion of the theory of Abelian difference sets, see also [[#References|[a1]]], Chap. VI.
9 KB (1,331 words) - 19:36, 13 February 2024
...in a certain sense, a measure of their deviation from being true. Similar generalizations of the structure theorems of linear algebra appear in topology. A vector sp
...jects may be studied with the aid of the homotopy theory of vector bundles and of topological
14 KB (2,405 words) - 22:14, 10 January 2015
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...nifold]] of constant negative curvature, or, more generally, of a discrete group $ G $
8 KB (1,145 words) - 22:11, 5 June 2020
...ncyclopedia/old_files/data/A011/A.0101980 Almost\AAhperiodic function on a group
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12 KB (1,716 words) - 11:05, 10 May 2020
...ms is retained — associativity; this is the explanation of the term "semi-group" . Semi-groups are called ''monoids'' if they have, in addition, an identit
...ntial geometry (semi-groups of partial transformations), and the algebraic theory of automata (semi-groups of automata).
17 KB (2,435 words) - 09:18, 2 April 2023
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...es whose geometry was initially developed for the case of three dimensions and only later was generalized to a dimension $ n > 3 $;
12 KB (1,820 words) - 02:24, 21 January 2022
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and $ y a = b $
16 KB (2,475 words) - 19:16, 18 January 2024
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''of a [[Topological group|topological group]]''
24 KB (3,516 words) - 08:27, 6 June 2020
...e ring $R$ that is a topological space in which the operations of addition and multiplication, as well as the mapping $R\times A\to A$ ($(r,a)\to ra$), ar
...ctures, i.e. groups, semi-groups, rings, lattices, vector spaces, modules, and others, equipped with topologies in which the algebraic operations consider
9 KB (1,142 words) - 17:15, 15 November 2023
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...inear mappings), and linear, bilinear and quadratic functions (functionals and forms) on vector spaces.
9 KB (1,394 words) - 08:15, 9 January 2024
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...nite automata are supplemented by studies of their various generalizations and modifications which reflect various features of real systems. In the case o
18 KB (2,794 words) - 10:01, 25 April 2020
...t alphabet of pattern symbols or "variables". Elements of the [[Free semi-group|free semigroup]] of non-empty words $E^+$ are ''patterns''. For a pattern
...$p$ is $k$-unavoidable if it is unavoidable on every alphabet of size $k$ and correspondingly $k$-avoidable if it is avoidable on an alphabet of size $k$
3 KB (454 words) - 20:18, 7 November 2023
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and where
6 KB (908 words) - 19:48, 12 January 2024
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in which each row and each column are permutations of the elements of a finite set $ S $
11 KB (1,681 words) - 11:23, 17 March 2023
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...ere used in [[#References|[a7]]] to investigate groups given by generators and relations.
10 KB (1,435 words) - 07:01, 29 March 2024
$#C+1 = 125 : ~/encyclopedia/old_files/data/K055/K.0505520 Kleinian group
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13 KB (1,863 words) - 19:27, 17 January 2024
...that is invariant under some [[Discrete group of transformations|discrete group of transformations]] $ \Gamma $ of analytic transformations of a given c
...al complex space $ \mathbf C ^{n} $ that are invariant under a discrete group $ \Gamma $ of automorphisms of this domain.
17 KB (2,502 words) - 06:16, 12 July 2022
...://www.encyclopediaofmath.org/legacyimages/m/m062/m062160/m0621601.png" />
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...www.encyclopediaofmath.org/legacyimages/m/m062/m062160/m0621608.png" /> if
and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofma
34 KB (4,631 words) - 18:28, 30 November 2016
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are the fields of components of the velocity, of the pressure and of the temperature of a turbulent fluid flow (see [[#References|[1]]]). An
9 KB (1,319 words) - 08:09, 6 June 2020
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that is graded by means of an Abelian group $ A $,
20 KB (2,828 words) - 19:11, 26 March 2023
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and $ \delta $
8 KB (1,166 words) - 19:43, 5 June 2020
...among the widely available systems is REDUCE, which runs on many platforms and is still being further developed.
...1983. Maple was released in 1986, while Scratchpad II (later called AXIOM) and Mathematica appeared by 1988.
9 KB (1,292 words) - 20:51, 18 September 2016
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...ntribution from 1929 [[#References|[a7]]] (see also [[#References|[a8]]]), and Wolff's boundary version of the [[Schwarz lemma|Schwarz lemma]] from 1926 [
14 KB (2,118 words) - 16:09, 11 February 2024
...$A$ is called a left polygon over a monoid $R$ if for any $\lambda \in R$ and $a \in A$ the product $\lambda a \in A$ is defined, such that
and
6 KB (1,055 words) - 05:59, 22 April 2023