[[Semi-group|semi-group]]. In particular, if $S$ is a group, one obtains a
...p $S$ is semi-simple if and only if all linear representations of the semi-group $S$ over the algebra $\Phi$ are reducible.
2 KB (344 words) - 19:41, 8 December 2015
...7 : ~/encyclopedia/old_files/data/P071/P.0701010 \BMI \Gpi\EMI\AAhsolvable group
A generalization of the concept of a [[Solvable group|solvable group]]. Let $ \pi $
3 KB (420 words) - 08:06, 6 June 2020
A ''torsion group'' (also called ''periodic group'')
[[Group|group]] in which every element has finite order. Any torsion
1 KB (170 words) - 21:20, 29 April 2012
...on, that is, it can be imbedded in a unique locally nilpotent torsion-free group $G^*$ such that all equations of the form $x^n=g$ are solvable in $G^*$, wh
<TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian
1 KB (227 words) - 18:28, 5 April 2018
...://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c1104002.png" />-
group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org
...ges/c/c110/c110400/c11040010.png" /> is partially ordered with the induced
order: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/
15 KB (2,061 words) - 17:13, 7 February 2011
''totally ordered group''
A [[Po-group| $ po $-
4 KB (587 words) - 08:03, 6 June 2020
...eal numbers in the interval $(0,1)$ and the symbol $\infty$ with the usual order and with the operations:
The former case occurs if and only if $S$ is a [[semi-group with cancellation]].
3 KB (413 words) - 09:17, 2 April 2023
...egacyimages/h/h110/h110290/h11029025.png" />, which is identified with the
group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
....encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029051.png" />-Hopf
order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
13 KB (1,801 words) - 19:17, 12 January 2018
...rs from the theory of groups (i.e. $2$-groups). Thus, if $n\geq 3$, an $n$-group has no analogue of the unit element.
Let $\Gamma(\circ)$ be a group with multiplication operation $\circ$; let $n\geq 3$ be an arbitrary intege
2 KB (307 words) - 21:09, 21 November 2014
...with the operation of multiplication in the skew-field. The multiplicative group of a field is Abelian.
...lic group or is a finitely-solvable group and has an extension to a linear group over a skew-field. Some cases are known, e.g., [[#References|[a2]]].
1 KB (219 words) - 20:48, 21 December 2014
''in a group $G$''
...he order of a subgroup $H$ by its index $\left|G:H\right|$ is equal to the order of $G$ (Lagrange's theorem). This relationship applies to finite groups $G$
1 KB (186 words) - 11:18, 20 April 2012
...ng all elements of the groupoid (in any order); these symbols (in the same order) are also listed in front of the first column. If the groupoid has an ident
Cayley tables were first used by A. Cayley in 1854 for groups (cf. [[Group|Group]]).
2 KB (300 words) - 16:43, 4 June 2020
''of a finite group $ G $''
see [[Sylow subgroup|Sylow subgroup]]). A group $ G $
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be a [[Finite group|finite group]] and $ \pi $
a subset of the prime numbers that divide the order $ n $
2 KB (287 words) - 08:24, 6 June 2020
$#C+1 = 18 : ~/encyclopedia/old_files/data/P074/P.0704570 Primitive group of permutations,
''primitive permutation group''
3 KB (484 words) - 08:07, 6 June 2020
...g/legacyimages/f/f120/f120130/f1201301.png" /> denote an arbitrary [[
Group|
group]] (finite or infinite) and let <img align="absmiddle" border="0" src="https
...opediaofmath.org/legacyimages/f/f120/f120130/f12013038.png" /> is a finite
group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/
16 KB (2,143 words) - 17:10, 7 February 2011
The ''complexification of a Lie group $G$ over $\R$'' is
a complex Lie group $G_\C$ containing $G$ as a real Lie subgroup such that the Lie algebra $\de
3 KB (501 words) - 15:05, 13 April 2024
''right-ordered group''
A [[Group|group]] $ G $
4 KB (580 words) - 06:18, 28 March 2023
...n group theory. Most are not intrinsic to a group itself, but pertain to a group acting on something.
==Regular group of permutations.==
4 KB (716 words) - 07:52, 25 November 2023
$#C+1 = 13 : ~/encyclopedia/old_files/data/L058/L.0508600 Lie group, Banach
endowed with a group structure and an analytic Banach manifold structure (cf. [[Banach analytic
3 KB (427 words) - 22:16, 5 June 2020