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.
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...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
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...). 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]]).
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''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 $S$.
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''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,
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...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]]
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...|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 $
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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