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  • 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 <math>e</math> 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) - 03:15, 16 April 2017
  • Let <math>G</math> be a [[group]]. The '''Frattini subgroup''' of <math>G</math>, denoted <math>\Phi(G)</ma When <math>G</math> 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]] <math>H</math> of a [[group]] <math>G</math> is termed a fully invariant subgroup of <math>G</math> if
    17 KB (2,241 words) - 21:58, 30 May 2020
  • A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is said to be '''isomorph-free''' if it satisfies the foll # <math>H</math> is a [[co-Hopfian group]], and whenever <math>K \le G</math> such that <math>H \cong K</math>, then
    11 KB (1,293 words) - 02:34, 20 April 2016
  • {{particular group}} The Klein four-group, usually denoted <math>V_4</math>, 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 <math>H</math> of a group <math>G</math> 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 <math>G</math> is termed nilpotent if ... ...its [[upper central series]] stabilizes after a finite length at the whole group || there is a nonnegative integer <math>c</math> such that <math>Z^c(G) = G
    18 KB (2,458 words) - 23:24, 9 September 2016
  • ...n whole group are conjugate in intermediate subgroups, conjugates in whole group are conjugate in join}} ...ronormal if... (right-action convention) !! A subgroup <math>H</math> of a group <math>G</math> is pronormal if... (left-action convention)
    20 KB (2,702 words) - 04:06, 10 November 2014
  • ...up of a group is termed a retract if ... !! A subgroup <math>H</math> of a group <math>G</math> is termed a retract of <math>G</math> 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 <math>G</math> 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 <math>S_3</math> 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 <math>\operatorname{Sym}(4)</math>, 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 <math>*</math> is the [[composition operator]] on subgroup properties, then a property <math>p</math> is transitive if <math>p * p \le p</math>. ...group property is t.i. if it is transitive, and is always satisfied by any group as a subgroup of itself. || || ||
    7 KB (1,012 words) - 15:02, 1 June 2020
  • {{pivotal group property}} The term '''abelian group''' comes from Niels Henrick Abel, a mathematician who worked with groups ev
    13 KB (1,912 words) - 15:35, 11 April 2017
  • ...[[group]] is characteristic in it if ... !! A subgroup <math>H</math> of a group <math>G</math> is called a characteristic subgroup of <math>G</math> if ... | 1 || automorphism-invariant || every [[automorphism]] of the whole group takes the subgroup to within itself. || for every [[automorphism]] <math>\v
    40 KB (4,935 words) - 13:17, 2 June 2020
  • ...itions (except the first one, as noted) assumes that we ''already'' have a group and a subgroup. To prove normality using any of these definitions, we first ...p]] of a [[group]] is normal in it if... !! A subgroup <math>H</math> of a group <math>G</math> is normal in <math>G</math> if ... !! Applications to... !!
    43 KB (5,764 words) - 13:39, 2 August 2018
  • ...avariant if... (right-action convention) !! A subgroup <math>H</math> of a group <math>G</math> is termed automorph-conjugate or intravariant if... (left-ac ...(i.e. any subgroup to which it can go via an [[automorphism]] of the whole group), is also [[Defining ingredient::conjugate subgroups|conjugate]] to the sub
    14 KB (1,758 words) - 17:39, 21 December 2014

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