# 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. | ||

+ | |||

+ | ==Arithmetic functions== | ||

+ | |||

+ | ===Basic arithmetic functions=== | ||

+ | |||

+ | {{compare and contrast arithmetic functions|order = 2520}} | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Function !! Value !! Similar groups !! Explanation | ||

+ | |- | ||

+ | | {{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> | ||

+ | |- | ||

+ | | {{arithmetic function given order|exponent of a group|420|2520}} || | ||

+ | |- | ||

+ | | [[derived length]] || -- || || not a [[solvable group]] | ||

+ | |- | ||

+ | | [[nilpotency class]] || -- || || not a [[nilpotent group]] | ||

+ | |- | ||

+ | | {{arithmetic function value given order|Frattini length|1|2520}} || [[Frattini-free group]]: intersection of all maximal subgroups is trivial | ||

+ | |- | ||

+ | | {{arithmetic function value given order|minimum size of generating set|2|2520}} || | ||

+ | |} | ||

+ | |||

+ | ===Arithmetic functions of a counting nature=== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Function !! Value !! Explanation | ||

+ | |- | ||

+ | | [[number of subgroups]] || [[arithmetic function value::number of subgroups;3786|3786]] || See [[subgroup structure of alternating group:A7]], [[subgroup structure of alternating groups]] | ||

+ | |- | ||

+ | | [[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]] | ||

+ | |- | ||

+ | | [[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]] | ||

+ | |} |

## 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 groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

This group is defined as the alternating group of degree , i.e., the alternating group on a set of size . 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 : |

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 |