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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[/itex]. Then, $H$ 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 $p$-group with cyclic maximal subgroup is a finite $p$-group (i.e., a [[group of prime power order]] wh
344 bytes (63 words) - 21:44, 9 July 2008
• This page classifies all finite $p$-groups that possess a cyclic maximal subgroup. ...ime, and $G$ be a group of order $p^n$ with a cyclic maximal subgroup $M$ (so $M \cong \mathbb{Z}/p^{n-1}\mathbb{Z}$). Then
15 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 $G$ be a [[finite solvable group]] and $M$ be a [[maximal subgroup]] of $G$. Then, the [[index]] $[G:M]$ is a power of a
531 bytes (68 words) - 12:18, 28 September 2008
• Suppose $G$ is a [[primitive group]] with core-free maximal subgroup $M$. Then, if $G$ 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 $K \le G$ is a [[maximal subgroup]] that is also a [[characteristic subgroup]] (in other words, $K</math 2 KB (260 words) - 14:33, 22 September 2008 • ...denotes the [[Frattini subgroup]] of [itex]P$) and such that every [[maximal subgroup]] of $P$ 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 $G$ is a [[primitive group]], $M$ is a [[core-free maximal subgroup]] of $G$, and $N$ is a [[minimal normal subgroup]] of
2 KB (322 words) - 05:32, 24 January 2015
• weaker = group in which every maximal subgroup is normal}} Then, any [[maximal subgroup]] of $G$ is a [[normal subgroup]].
1 KB (186 words) - 01:31, 7 February 2009
• ...[prime number]]. A subgroup $H$ of $G$ is termed a ''maximal subgroup of $p$-Sylow subgroup''' if it satisfies the following equivalen * $H$ is a [[maximal subgroup]] of some $p$-[[Sylow subgroup]] of $G$.
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 categ
2 KB (315 words) - 22:42, 7 May 2008
• ...[/itex], $|P \cap C| = |Q \cap C|$. Then, $P$ is a [[maximal subgroup]] of $Sym(n)$ if and only if $Q$ 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 is
2 KB (387 words) - 23:24, 7 May 2008
• * if $M$ is a [[maximal subgroup]] of $G$ and $y \in M$ has prime power order, then th
876 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]] -- $\l | sign, kernel a cyclic maximal subgroup || 2 || 1 || 1 || any || [[Z8 in M16]] -- either [itex]\langle a \rangle</m 13 KB (1,733 words) - 00:16, 4 June 2012 • * [[Nilpotent implies every maximal subgroup is normal]] * The maximal subgroup of [itex]P$ all contain $\Phi(P)$, the [[Frattini subgroup
11 KB (1,600 words) - 23:24, 3 November 2010
• ...oup is normal]]. The upshot is that for a finite $p$-group, any maximal subgroup is normal and has index $p$. | PA2 || There exists a maximal subgroup $Q$ of $P$ containing $A$. $Q$
13 KB (2,235 words) - 22:24, 29 September 2010
• # Every [[maximal subgroup]] is [[normal subgroup|normal]]
5 KB (655 words) - 04:11, 16 April 2017
• ...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 w
3 KB (441 words) - 23:31, 7 May 2008
• ...enerally, when every proper subgroup of $G$ is contained in a [[maximal subgroup]], then this condition is equivalent to saying that $N$ is conta
1 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 characteri
4 KB (566 words) - 16:32, 20 April 2017
• ...intersection of all maximal subgroups, biggest subgroup contained in every maximal subgroup}} ...section]] of all subgroups $M \le G$, where $M$ is [[maximal subgroup|maximal]] in $G$
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 $H = \Phi(G)$ be its [[Frattini subgroup]]. Then, $5 KB (744 words) - 23:32, 7 May 2008 • ...[[finite group]] and [itex]\Phi(G)$ denote the intersection of all [[maximal subgroup]]s of $G$ (the so-called [[Frattini subgroup]] of $G$ ...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, $\Phi(N)$, the [[Frattini subgroup]] of $N$,
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 additi
3 KB (613 words) - 23:33, 7 May 2008
• * Group in which every maximal subgroup is normal
498 bytes (56 words) - 23:43, 7 May 2008
• * The [[Frattini subgroup]] is the intersection of all [[maximal subgroup]]s
3 KB (457 words) - 23:43, 7 May 2008

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