# Search results

• An element of a [[group]] is termed central if the following equivalent conditions hold: # It commutes with every element of the group
15 KB (2,081 words) - 20:14, 1 June 2016
• {{particular group}} ...times the ''octic group'', is defined by the following [[presentation of a group|presentation]], with [itex]e[/itex] denoting the identity element:
19 KB (2,660 words) - 13:15, 14 February 2015
• A [[group]] is said to be '''finitely generated''' if it satisfies the following equi # Every [[generating set]] of the group has a subset that is finite and is also a generating set.
7 KB (920 words) - 04:32, 16 January 2022
• Let [itex]G[/itex] be a [[group]]. The '''Frattini subgroup''' of [itex]G[/itex], denoted [itex]\Phi(G)</ma When [itex]G[/itex] is a [[group in which every subgroup is contained in a maximal subgroup]], then the Frat
4 KB (603 words) - 06:49, 20 April 2017
• A [[subgroup]] of a [[group]] is termed '''fully invariant''' or '''fully characteristic''' if it satis ...] is termed fully invariant if ... !! A [[subgroup]] [itex]H[/itex] of a [[group]] [itex]G[/itex] is termed a fully invariant subgroup of [itex]G[/itex] if
17 KB (2,242 words) - 15:35, 18 August 2021
• A [[subgroup]] [itex]H[/itex] of a [[group]] [itex]G[/itex] is said to be '''isomorph-free''' if it satisfies the foll # [itex]H[/itex] is a [[co-Hopfian group]], and whenever [itex]K \le G[/itex] such that [itex]H \cong K[/itex], then
11 KB (1,293 words) - 02:34, 20 April 2016
• {{particular group}} The Klein four-group, usually denoted [itex]V_4[/itex], is defined in the following equivalent w
5 KB (643 words) - 16:32, 21 December 2014
• A nontrivial subgroup of a [[group]] is termed a '''minimal normal subgroup''' if it is [[normal subgroup|norm A nontrivial subgroup [itex]H[/itex] of a group [itex]G[/itex] is termed a '''minimal normal subgroup''' if it is normal an
6 KB (941 words) - 05:30, 24 January 2015
• ! No. !! Shorthand !! A group is termed nilpotent if ... !! A group [itex]G[/itex] is termed nilpotent if ... ...its [[upper central series]] stabilizes after a finite length at the whole group || there is a nonnegative integer [itex]c[/itex] such that [itex]Z^c(G) = G
18 KB (2,561 words) - 02:11, 27 December 2021
• ...n whole group are conjugate in intermediate subgroups, conjugates in whole group are conjugate in join}} ...ronormal if... (right-action convention) !! A subgroup [itex]H[/itex] of a group [itex]G[/itex] is pronormal if... (left-action convention)
20 KB (2,702 words) - 04:06, 10 November 2014
• | [[Direct product of group of rational numbers and group of rational numbers modulo integers]] || It said [itex]\mathbb{Q} *_{\mathb | [[Nilpotent group]] || One of the shorthands didn't correspond to details, and the verbal sta
61 KB (8,704 words) - 03:36, 27 December 2021
• ...up of a group is termed a retract if ... !! A subgroup [itex]H[/itex] of a group [itex]G[/itex] is termed a retract of [itex]G[/itex] if ... ...idempotent endomorphism || there is an idempotent [[endomorphism]] of the group whose image is precisely that subgroup. This idempotent endomorphism is ter
7 KB (995 words) - 03:55, 9 March 2020
• ...number among groups, group without any proper nontrivial normal subgroup, group without any proper nontrivial quotients}} ! No. !! Shorthand !! A group is simple if ... !! A group [itex]G[/itex] is simple if ...
10 KB (1,387 words) - 20:01, 23 April 2014
• A [[subgroup]] of a [[group]] is termed '''subnormal''' if any of the following equivalent conditions h ...ing chain of subgroups starting from the subgroup and going till the whole group, such that each is a [[defining ingredient::normal subgroup]] of its succes
18 KB (2,510 words) - 15:40, 16 April 2017
• The symmetric group [itex]S_3[/itex] can be defined in the following equivalent ways: ...amily::symmetric group of prime degree]] and [[member of family::symmetric group of prime power degree]].
18 KB (2,642 words) - 20:52, 26 January 2020
• {{particular group}} .../math> or [itex]\operatorname{Sym}(4)[/itex], also termed the '''symmetric group of degree four''', is defined in the following equivalent ways:
16 KB (2,246 words) - 20:05, 26 January 2020
• If [itex]*[/itex] is the [[composition operator]] on subgroup properties, then a property [itex]p[/itex] is transitive if [itex]p * p \le p[/itex]. ...group property is t.i. if it is transitive, and is always satisfied by any group as a subgroup of itself. || || ||
7 KB (1,022 words) - 13:44, 1 October 2021
• {{particular group}} ...he group with only one element, which is its identity element. The trivial group is usually denoted as [itex]1[/itex], [itex]\{ 1 \}[/itex], or [itex]\{ e \
3 KB (389 words) - 19:44, 31 March 2022
• {{pivotal group property}} The term '''abelian group''' comes from Niels Henrick Abel, a mathematician who worked with groups ev
13 KB (1,904 words) - 19:41, 31 March 2022
• ...[[group]] is characteristic in it if ... !! A subgroup [itex]H[/itex] of a group [itex]G[/itex] is called a characteristic subgroup of [itex]G[/itex] if ... | 1 || automorphism-invariant || every [[automorphism]] of the whole group takes the subgroup to within itself. || for every [[automorphism]] [itex]\v
41 KB (4,989 words) - 13:56, 8 June 2020

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