# Difference between revisions of "Automorphism group of alternating group:A6"

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# It is the [[member of family::projective semilinear group]] of [[member of family::projective semilinear group of degree two|degree two]] over [[field:F9|the field of nine elements]], i.e., it is the group <math>P\Gamma L(2,9)</math>. | # It is the [[member of family::projective semilinear group]] of [[member of family::projective semilinear group of degree two|degree two]] over [[field:F9|the field of nine elements]], i.e., it is the group <math>P\Gamma L(2,9)</math>. | ||

− | Note that for any <math>n \ne 2,3,6</math>, the [[automorphism group]] of the alternating group <math>A_n</math> is precisely the symmetric group <math>S_n</math>, which is a [[complete group]]. The case <math>n = 2</math> is uninteresting. For <math>n = 3</math>, the automorphism group of <math>A_n</math> is <math> | + | Note that for any <math>n \ne 2,3,6</math>, the [[automorphism group]] of the alternating group <math>A_n</math> is precisely the symmetric group <math>S_n</math>, which is a [[complete group]]. The case <math>n = 2</math> is uninteresting. For <math>n = 3</math>, the automorphism group of <math>A_n</math> is <math>\mathbb{Z}/2\mathbb{Z}</math>, the [[cyclic group:Z2|cyclic group of order two]]. The case <math>n = 6</math> is the only case where the automorphism group of the alternating group is strictly ''bigger'' than the symmetric group. Similarly, it is the only case where the automorphism group of the symmetric group is strictly ''bigger'' than the symmetric group. {{further|[[symmetric groups on finite sets are complete]]}} |

==Arithmetic functions== | ==Arithmetic functions== |

## Latest revision as of 01:11, 7 June 2015

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 in the following equivalent ways:

- It is the automorphism group of alternating group:A6.
- It is the automorphism group of symmetric group:S6.
- It is the projective semilinear group of degree two over the field of nine elements, i.e., it is the group .

Note that for any , the automorphism group of the alternating group is precisely the symmetric group , which is a complete group. The case is uninteresting. For , the automorphism group of is , the cyclic group of order two. The case is the only case where the automorphism group of the alternating group is strictly *bigger* than the symmetric group. Similarly, it is the only case where the automorphism group of the symmetric group is strictly *bigger* than the symmetric group. `Further information: symmetric groups on finite sets are complete`

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 1440#Arithmetic functions

### Basic arithmetic functions

Function | Value | Similar groups | Explanation for function value |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 1440 | groups with same order | As : As : As : |

### Arithmetic functions of a counting nature

Function | Value | Similar groups | Explanation for function value |
---|---|---|---|

number of conjugacy classes | 13 | groups with same order and number of conjugacy classes | groups with same number of conjugacy classes | As (odd): (more here) |

## GAP implementation

### Group ID

This finite group has order 1440 and has ID 5841 among the groups of order 1440 in GAP's SmallGroup library. For context, there are 5,958 groups of order 1440. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(1440,5841)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(1440,5841);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [1440,5841]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description | Functions used |
---|---|

AutomorphismGroup(AlternatingGroup(6)) |
AutomorphismGroup, AlternatingGroup |

AutomorphismGroup(SymmetricGroup(6)) |
AutomorphismGroup, SymmetricGroup |