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**Create the page ""maximal+subgroup"" on this wiki!** See also the search results found.

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- The notion of maximal subgroup probably dates back to the very beginning of group theory. A '''maximal subgroup''' of a group is defined in the following equivalent ways:4 KB (533 words) - 22:48, 21 November 2008
- A '''group in which every maximal subgroup is normal''' is a group satisfying the following equivalent conditions: * Any [[maximal subgroup]] (i.e. any [[proper subgroup]] not contained in any other proper subgroup)2 KB (276 words) - 00:56, 26 December 2015
- stronger = maximal subgroup| ...s stronger than the property of being a [[fact about::group in which every maximal subgroup is normal]].2 KB (284 words) - 12:36, 19 August 2011
- #redirect [[Group in which every maximal subgroup is normal]]61 bytes (9 words) - 12:12, 29 May 2008
- ...mpleted subgroup]] is contained in a [[fact about::maximal subgroup;3| ]][[maximal subgroup]]. ...>\langle H, x \rangle = G</math>. Then, <math>H</math> is contained in a [[maximal subgroup]].666 bytes (110 words) - 23:51, 5 July 2019
- #redirect [[Group in which every proper subgroup is contained in a maximal subgroup]]85 bytes (13 words) - 18:19, 26 June 2008
- ...roup that is not the whole group) is contained in a [[defining ingredient::maximal subgroup]] (a proper subgroup not contained in any other proper subgroup).662 bytes (86 words) - 15:18, 13 July 2008
- A finite <math>p</math>-group with cyclic maximal subgroup is a finite <math>p</math>-group (i.e., a [[group of prime power order]] wh344 bytes (63 words) - 21:44, 9 July 2008
- This page classifies all finite <math>p</math>-groups that possess a cyclic maximal subgroup. ...ime, and <math>G</math> be a group of order <math>p^n</math> with a cyclic maximal subgroup <math>M</math> (so <math>M \cong \mathbb{Z}/p^{n-1}\mathbb{Z}</math>). Then15 KB (2,590 words) - 19:36, 25 July 2009
- #redirect [[Group in which every proper subgroup is contained in a maximal subgroup]]85 bytes (13 words) - 15:17, 13 July 2008
- Let <math>G</math> be a [[finite solvable group]] and <math>M</math> be a [[maximal subgroup]] of <math>G</math>. Then, the [[index]] <math>[G:M]</math> is a power of a531 bytes (68 words) - 12:18, 28 September 2008
- Suppose <math>G</math> is a [[primitive group]] with core-free maximal subgroup <math>M</math>. Then, if <math>G</math> has an Abelian [[minimal normal sub ===Maximal subgroup===3 KB (442 words) - 11:23, 2 January 2009
- #redirect [[Nilpotent implies every maximal subgroup is normal]]64 bytes (8 words) - 20:15, 21 September 2008
- 44 bytes (6 words) - 14:27, 22 September 2008
- ...ath> is [[central subgroup|central]]. Then, if <math>K \le G</math> is a [[maximal subgroup]] that is also a [[characteristic subgroup]] (in other words, <math>K</math2 KB (260 words) - 14:33, 22 September 2008
- ...denotes the [[Frattini subgroup]] of <math>P</math>) and such that every [[maximal subgroup]] of <math>P</math> is a [[characteristic subgroup]].773 bytes (128 words) - 23:41, 8 April 2009
- A '''group in which every maximal subgroup is characteristic''' is a group satisfying the following equivalent conditi * Any [[defining ingredient::maximal subgroup]] is a [[defining ingredient::characteristic subgroup]].1,019 bytes (131 words) - 22:47, 15 September 2009
- ...e <math>G</math> is a [[primitive group]], <math>M</math> is a [[core-free maximal subgroup]] of <math>G</math>, and <math>N</math> is a [[minimal normal subgroup]] of2 KB (322 words) - 05:32, 24 January 2015
- weaker = group in which every maximal subgroup is normal}} Then, any [[maximal subgroup]] of <math>G</math> is a [[normal subgroup]].1 KB (186 words) - 01:31, 7 February 2009
- ...[prime number]]. A subgroup <math>H</math> of <math>G</math> is termed a ''maximal subgroup of <math>p</math>-Sylow subgroup''' if it satisfies the following equivalen * <math>H</math> is a [[maximal subgroup]] of some <math>p</math>-[[Sylow subgroup]] of <math>G</math>.1 KB (184 words) - 22:47, 4 March 2009

## Page text matches

- ...s subgroup-defining function::Frattini subgroup]] || intersection of all [[maximal subgroup]]s ||1 KB (187 words) - 22:11, 28 June 2011
- ...subgroup property|subgroup properties]], such as [[subnormal subgroup]], [[maximal subgroup]] etc. A full list of subgroup properties is available at the monster categ2 KB (315 words) - 04:38, 2 May 2022
- ...</math>, <math>|P \cap C| = |Q \cap C|</math>. Then, <math>P</math> is a [[maximal subgroup]] of <math>Sym(n)</math> if and only if <math>Q</math> is.2 KB (246 words) - 23:14, 7 May 2008
- Every [[maximal subgroup]] is either normal or contranormal. * Non-normal [[maximal subgroup]]5 KB (676 words) - 22:50, 22 November 2008
- If a group has a [[maximal subgroup]] that is also core-free, then it is termed a [[primitive group]]. This is2 KB (387 words) - 23:24, 7 May 2008
- * if <math>M</math> is a [[maximal subgroup]] of <math>G</math> and <math>y \in M</math> has prime power order, then th876 bytes (150 words) - 23:24, 7 May 2008
- | sign, kernel a non-cyclic maximal subgroup || 1 || 1 || 1 || any || [[direct product of Z4 and Z2 in M16]] -- <math>\l | sign, kernel a cyclic maximal subgroup || 2 || 1 || 1 || any || [[Z8 in M16]] -- either <math>\langle a \rangle</m13 KB (1,733 words) - 00:16, 4 June 2012
- * [[Nilpotent implies every maximal subgroup is normal]] * The maximal subgroup of <math>P</math> all contain <math>\Phi(P)</math>, the [[Frattini subgroup11 KB (1,600 words) - 23:24, 3 November 2010
- ...oup is normal]]. The upshot is that for a finite <math>p</math>-group, any maximal subgroup is normal and has index <math>p</math>. | PA2 || There exists a maximal subgroup <math>Q</math> of <math>P</math> containing <math>A</math>. <math>Q</math>13 KB (2,235 words) - 22:24, 29 September 2010
- # Every [[maximal subgroup]] is [[normal subgroup|normal]].5 KB (655 words) - 15:52, 16 January 2022
- ...group satisfies the property that every proper subgroup is contained in a maximal subgroup. The proof of this monotonicity uses the statement of this article, along w3 KB (441 words) - 23:31, 7 May 2008
- ...enerally, when every proper subgroup of <math>G</math> is contained in a [[maximal subgroup]], then this condition is equivalent to saying that <math>N</math> is conta1 KB (145 words) - 21:16, 7 July 2008
- ...[[defining ingredient::Frattini subgroup]] (the intersection of all its [[maximal subgroup]]s) is [[trivial group|trivial]]. ...imple group]] or [[characteristically simple group]] that has at least one maximal subgroup, is Frattini-free. This is because the Frattini subgroup must be characteri4 KB (566 words) - 16:32, 20 April 2017
- ...intersection of all maximal subgroups, biggest subgroup contained in every maximal subgroup}} ...section]] of all subgroups <math>M \le G</math>, where <math>M</math> is [[maximal subgroup|maximal]] in <math>G</math>4 KB (603 words) - 06:49, 20 April 2017
- ...generally, of a group where every [[proper subgroup]] is contained in a [[maximal subgroup]])is an [[ACIC-group]]. ...oup]] (or more generally, a group where every subgroup is contained in a [[maximal subgroup]]), and <math>H = \Phi(G)</math> be its [[Frattini subgroup]]. Then, <math>5 KB (744 words) - 23:32, 7 May 2008
- ...[[finite group]] and <math>\Phi(G)</math> denote the intersection of all [[maximal subgroup]]s of <math>G</math> (the so-called [[Frattini subgroup]] of <math>G</math> ...tini subgroup]] of a (here, finite) group is the intersection of all its [[maximal subgroup]]s.2 KB (307 words) - 16:02, 18 July 2008
- ...mal subgroup is a [[group in which every proper subgroup is contained in a maximal subgroup]]. ...satisfies the property that every [[proper subgroup]] is contained in a [[maximal subgroup]]. Then, <math>\Phi(N)</math>, the [[Frattini subgroup]] of <math>N</math>,5 KB (751 words) - 13:25, 2 July 2008
- ...is that in the general structure tree, the index of the normal core of the maximal subgroup may not in general have small value. To ensure this, we need special additi3 KB (613 words) - 23:33, 7 May 2008
- * Group in which every maximal subgroup is normal498 bytes (56 words) - 23:43, 7 May 2008
- * The [[Frattini subgroup]] is the intersection of all [[maximal subgroup]]s3 KB (457 words) - 23:43, 7 May 2008