...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
Please remove this comment and the {{TEX|auto}} line below,
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
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...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...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.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
Please remove this comment and the {{TEX|auto}} line below,
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
Please remove this comment and the {{TEX|auto}} line below,
...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
Please remove this comment and the {{TEX|auto}} line below,
of a locally compact group $ G $
13 KB (1,974 words) - 22:12, 5 June 2020
Please remove this comment and the {{TEX|auto}} line below,
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