Subgroup structure of alternating group:A7 - Revision history

Vipul at 23:33, 18 April 2012

2012-04-18T23:33:28Z

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	\| [[normal subgroup]]s \|\| only the whole group and the trivial subgroup, because the group is simple. See [[alternating groups are simple]].		\| [[normal subgroup]]s \|\| only the whole group and the trivial subgroup, because the group is simple. See [[alternating groups are simple]].
	\|-		\|-
	\| subgroups that are [[simple non-abelian group]]s \|\| [[alternating group:A5]] (order 60), [[projective special linear group:PSL(3,2)]] (order 168, the embedding arises via the natural permutation representation on the two-dimensional projective space over [[field:F2]], which has size <math>2^2 + 2 + 1 = 7</math>), [[alternating group:A6]] (order 360)		\| subgroups that are [[simple non-abelian group]]s (apart from the whole group itself) \|\| [[alternating group:A5]] (order 60), [[projective special linear group:PSL(3,2)]] (order 168, the embedding arises via the natural permutation representation on the two-dimensional projective space over [[field:F2]], which has size <math>2^2 + 2 + 1 = 7</math>), [[alternating group:A6]] (order 360)
	\|}		\|}

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Vipul: /* Quick summary */

2012-04-18T23:30:35Z

Quick summary

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	\|-		\|-
	\| [[normal subgroup]]s \|\| only the whole group and the trivial subgroup, because the group is simple. See [[alternating groups are simple]].		\| [[normal subgroup]]s \|\| only the whole group and the trivial subgroup, because the group is simple. See [[alternating groups are simple]].
			\|-
			\| subgroups that are [[simple non-abelian group]]s \|\| [[alternating group:A5]] (order 60), [[projective special linear group:PSL(3,2)]] (order 168, the embedding arises via the natural permutation representation on the two-dimensional projective space over [[field:F2]], which has size <math>2^2 + 2 + 1 = 7</math>), [[alternating group:A6]] (order 360)
	\|}		\|}

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Vipul: /* Tables for quick information */

2012-04-18T23:27:19Z

Tables for quick information

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	\| [[normal subgroup]]s \|\| only the whole group and the trivial subgroup, because the group is simple. See [[alternating groups are simple]].		\| [[normal subgroup]]s \|\| only the whole group and the trivial subgroup, because the group is simple. See [[alternating groups are simple]].
	\|}		\|}

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Vipul at 23:26, 18 April 2012

2012-04-18T23:26:09Z

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	group = alternating group:A7\|		group = alternating group:A7\|
	connective = of}}		connective = of}}

			This article discusses the subgroup structure of [[alternating group:A7]], which is the [[alternating group]] on the set <math>\{ 1, 2,3,4,5,6,7\}</math>. The group has order 2520.

			==Family contexts==

			{\| class="sortable" border="1"
			! Family name !! Parameter values !! General discussion of subgroup structure of family
			\|-
			\| [[alternating group]] \|\| degree <math>n = 7</math>, i.e., the group <math>A_7</math> \|\| [[subgroup structure of alternating groups]]
			\|}
			==Tables for quick information==

			{{finite group subgroup structure facts to check against}}

			<section begin="summary"/>

			===Quick summary===

			{\| class="sortable" border="1"
			! Item !! Value
			\|-
			\| [[Number of subgroups]] \|\| 3786<br>Compared with <math>A_n, n = 3,4,5,\dots</math>: 2, 10, 59, 501, '''3786''', 48337, ...
			\|-
			\| [[Number of conjugacy classes of subgroups]] \|\| 40<br>Compared with <math>A_n, n = 3,4,5,\dots</math>: 2, 5, 9, 22, '''40''', 137, ...
			\|-
			\| [[Number of automorphism classes of subgroups]] \|\| 37<br>Compared with <math>A_n, n = 3,4,5,\dots</math>: 2, 5, 9, 16, '''37''', 112, ...
			\|-
			\| Isomorphism classes of [[Sylow subgroup]]s and the corresponding [[fusion system]]s \|\| 2-Sylow: [[dihedral group:D8]] (order 8) as [[D8 in A7]] (with its non-inner fusion system -- see [[fusion systems for dihedral group:D8]]). [[Sylow number]] is 315.<br>3-Sylow: [[elementary abelian group:E9]] (order 9) as [[E9 in A7]]. [[Sylow number]] is 70.<br>5-Sylow: [[cyclic group:Z5]] (order 5) as [[Z5 in A7]]. [[Sylow number]] is 126.<br>7-Sylow: [[cyclic group:Z7]] (order 7) as [[Z7 in A7]]. [[Sylow number]] is 120.
			\|-
			\| [[Hall subgroup]]s \|\| Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are <math>\{ 2,3 \}</math>-Hall subgroups (of order 72) and <math>\{ 2,3,5 \}</math>-Hall subgroups (of order 360), the latter being [[A6 in A7]]. Note that the <math>\{ 2,3 \}</math>-Hall subgroups are not contained in <math>\{ 2,3,5 \}</math>-Hall subgroups.
			\|-
			\| [[maximal subgroup]]s \|\| maximal subgroups have orders 72, 120, 168, 360.
			\|-
			\| [[normal subgroup]]s \|\| only the whole group and the trivial subgroup, because the group is simple. See [[alternating groups are simple]].
			\|}

Vipul: Created page with "{{group-specific information| information type = subgroup structure| group = alternating group:A7| connective = of}}"

2012-04-18T05:23:44Z

Created page with "{{group-specific information| information type = subgroup structure| group = alternating group:A7| connective = of}}"

New page

{{group-specific information|
information type = subgroup structure|
group = alternating group:A7|
connective = of}}