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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
  • ...he action is happening. For an irreducible representation <math>\varphi:G \to GL(n,K)</math> over a field <math>K</math>, the degree is <math>n</math>. ===Typical context: finite group and splitting field===
    15 KB (2,038 words) - 02:28, 28 May 2013
  • {{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
  • {{survey article|group}} ...ormal definition that we study these days in textbooks. The development of group theory beyond that point is not discussed here.
    13 KB (1,985 words) - 23:34, 15 March 2009
  • This article explores the various ways in which, given a group and a subgroup (through some kind of description) we can try proving that t * [[Replacing a subgroup by a normal subgroup]]: There are many techniques to guarantee, from the existence of a subgroup satisfying certain conditions,
    17 KB (2,706 words) - 02:55, 20 February 2013
  • {{group description rule}} A '''presentation''' of a group is the following data:
    8 KB (1,388 words) - 16:08, 31 December 2018
  • ...us, does ''not'' require login, but you can optionally provide an email ID to be notified of the fix). As we try to make Groupprops more reliable, this log is intended to keep a record of errors that satisfy ''either'' of these two criteria:
    60 KB (8,443 words) - 13:46, 1 October 2021
  • ...r''' of a group <math>G</math>, denoted <math>M(G)</math>, is an [[abelian group]] defined in the following equivalent ways: ...| It is the second [[defining ingredient::homology group for trivial group action]] <math>\! H_2(G;\mathbb{Z})</math>.
    8 KB (1,172 words) - 18:29, 29 June 2013
  • The property of being a subgroup whose index in the whole group is two, is stronger than the property of being a [[normal subgroup]]. ...n a [[finitary symmetric group]], the corresponding [[finitary alternating group]] is normal.
    6 KB (931 words) - 17:06, 12 September 2011
  • {{group property}} The '''symmetric group''' on a set is defined as follows:
    14 KB (2,163 words) - 00:40, 26 July 2011
  • {{subgroup property related to|geometric group theory}} {{subgroup property related to|combinatorial group theory}}
    10 KB (1,445 words) - 17:20, 26 July 2013
  • ...subgroup. To prove normality using any of these definitions, we first need to check that we actually have a subgroup. ...a group <math>G</math> is normal in <math>G</math> if ... !! Applications to... !! Additional comments
    43 KB (5,777 words) - 02:26, 20 August 2021
  • {{group-specific information| group = symmetric group:S3|
    33 KB (4,421 words) - 01:36, 27 February 2014
  • first = dihedral group:D8| second = quaternion group}}
    5 KB (737 words) - 14:04, 7 September 2011
  • ...]] with the binary operation and identity element inherited from the whole group). ==Related facts==
    8 KB (1,335 words) - 18:22, 2 July 2013
  • {{quotation|A sequel to this article, describing more advanced approaches to deducing facts about Sylow subgroups and Hall subgroups, is available at [[ ...s to prove that certain groups are not simple, refer [[using Sylow numbers to prove the existence of proper nontrivial normal subgroups]]}}
    26 KB (4,027 words) - 15:15, 12 April 2009
  • ...<math>p</math>-subgroup of <math>G</math> is a <math>p</math>-constrained group. ...ath>, not in <math>G</math>) and further, <math>A</math> has [[rank of a p-group|rank]] at least three. In other words, any generating set for <math>A</math
    9 KB (1,400 words) - 06:00, 30 July 2013
  • ...eover, <math>h</math> and <math>g</math> have the same element-wise action on <math>A</math>. ==Related facts==
    3 KB (553 words) - 16:15, 31 July 2009
  • {{odd-order-only p-group statement}} ===Hands-on statement===
    6 KB (896 words) - 20:45, 20 September 2009
  • ===In terms of subgroups=== ....e., there is no [[automorphism]] of <math>G</math> sending <math>H</math> to <math>K</math>).
    6 KB (993 words) - 21:51, 5 July 2019

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