Difference between revisions of "Alternating group:A7"

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This group is defined as the [[member of family::alternating group]] of degree <math>7</math>, i.e., the alternating group on a set of size <math>7</math>. In other words, it is the subgroup of [[symmetric group:S7]] comprising the [[even permutation]]s.
 
This group is defined as the [[member of family::alternating group]] of degree <math>7</math>, i.e., the alternating group on a set of size <math>7</math>. In other words, it is the subgroup of [[symmetric group:S7]] comprising the [[even permutation]]s.
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==Arithmetic functions==
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===Basic arithmetic functions===
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{{compare and contrast arithmetic functions|order = 2520}}
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{| class="sortable" border="1"
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! Function !! Value !! Similar groups !! Explanation
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|-
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| {{arithmetic function value order|2520}} || As alternating group <math>A_n, n = 7</math>: <math>n!/2 = 7!/2 = = (7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)/2 = 2520</math>
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|-
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| {{arithmetic function given order|exponent of a group|420|2520}} ||
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|-
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| [[derived length]] || -- || || not a [[solvable group]]
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|-
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| [[nilpotency class]] || -- || || not a [[nilpotent group]]
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|-
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| {{arithmetic function value given order|Frattini length|1|2520}} || [[Frattini-free group]]: intersection of all maximal subgroups is trivial
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|-
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| {{arithmetic function value given order|minimum size of generating set|2|2520}} ||
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|}
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===Arithmetic functions of a counting nature===
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{| class="sortable" border="1"
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! Function !! Value !! Explanation
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|-
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| [[number of subgroups]] || [[arithmetic function value::number of subgroups;3786|3786]] || See [[subgroup structure of alternating group:A7]], [[subgroup structure of alternating groups]]
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|-
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| [[number of conjugacy classes]] || [[arithmetic function value::number of conjugacy classes;9|9]] || See [[element structure of alternating group:A7]], [[element structure of alternating groups]]
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|-
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| [[number of conjugacy classes of subgroups]] || [[arithmetic function value::number of conjugacy classes of subgroups;40|40]] || See [[subgroup structure of alternating group:A7]], [[subgroup structure of alternating groups]]
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|}

Revision as of 02:24, 21 March 2012

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Definition

This group is defined as the alternating group of degree 7, i.e., the alternating group on a set of size 7. In other words, it is the subgroup of symmetric group:S7 comprising the even permutations.

Arithmetic functions

Basic arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 2520#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 2520 groups with same order As alternating group A_n, n = 7: n!/2 = 7!/2 = = (7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)/2 = 2520
Template:Arithmetic function given order
derived length -- not a solvable group
nilpotency class -- not a nilpotent group
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length Frattini-free group: intersection of all maximal subgroups is trivial
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set

Arithmetic functions of a counting nature

Function Value Explanation
number of subgroups 3786 See subgroup structure of alternating group:A7, subgroup structure of alternating groups
number of conjugacy classes 9 See element structure of alternating group:A7, element structure of alternating groups
number of conjugacy classes of subgroups 40 See subgroup structure of alternating group:A7, subgroup structure of alternating groups