# Difference between revisions of "Symmetric group:S5"

(→Elements) |
(→Arithmetic functions) |
||

Line 12: | Line 12: | ||

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

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

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

|- | |- | ||

− | | | + | | {{arithmetic function value order|120}} || as <math>\! S_k, k = 5:</math> <math>\! 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120</math> |

|- | |- | ||

− | + | { {{arithmetic function value exponent given order|60|120}} || Elements of order <math>2,3,4,5</math>. | |

|- | |- | ||

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

|- | |- | ||

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

|- | |- | ||

− | | | + | | {{arithmetic function value given order|Frattini length|1|120}} || [[Frattini-free group]]: intersection of maximal subgroups is trivial. |

|- | |- | ||

− | | | + | | {{arithmetic function value given order|minimum size of generating set|2|120}} || <math>(1,2), (1,2,3,4,5)</math>; see also [[symmetric group on a finite set is 2-generated]] |

|- | |- | ||

− | | | + | | {{arithmetic function value given order|subgroup rank of a group|2|120}} || |

|- | |- | ||

− | | | + | | {{arithmetic function value given order|max-length of a group|5|120}} || |

|- | |- | ||

− | | | + | | {{arithmetic function value given order|number of subgroups|156|120}} || |

|- | |- | ||

− | | | + | | {{arithmetic function value given order|number of conjugacy classes|7|120}} || as <math>S_k, k = 5</math>: the number of conjugacy classes is <math>p(k) = p(5) = 7</math>, where <math>p</math> is the [[number of unordered integer partitions]]. See also [[cycle type determines conjugacy class]] |

|- | |- | ||

− | | | + | | {{arithmetic function value given order|number of conjugacy classes of subgroups|19|120}} || |

|} | |} | ||

## Revision as of 16:13, 3 July 2010

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

The symmetric group is defined in the following equivalent ways:

- It is the group of all permutations on a set of five elements, i.e., it is the symmetric group of degree five. In particular, it is a symmetric group of prime degree and symmetric group of prime power degree.
- It is the projective general linear group of degree two over the field of five elements, i.e., .

### Presentation

## Arithmetic functions

{ exponent || 60 || groups with same order and exponent | groups with same exponent || Elements of order .## Group properties

COMPARE AND CONTRAST:Want to know more about how this group compares with symmetric groups of other degrees? Read contrasting symmetric groups of various degrees.

Property | Satisfied | Explanation | Comment |
---|---|---|---|

Abelian group | No | , don't commute | is non-abelian, . |

Nilpotent group | No | Centerless: The center is trivial | is non-nilpotent, . |

Metacyclic group | No | No cyclic normal subgroup | is not metacyclic, . |

Supersolvable group | No | No cyclic normal subgroup | is not supersolvable, . |

Solvable group | No | The subgroup is simple non-abelian | is simple and hence not solvable, . |

T-group | Yes | ||

HN-group | Yes | ||

Complete group | Yes | Centerless and every automorphism's inner | Symmetric groups are complete except the ones of degree . |

Monolithic group | Yes | Monolith is the alternating group | All symmetric groups are monolithic; is the only case the monolith is not the alternating group. |

One-headed group | Yes | The alternating group is the unique maximal normal subgroup | True for all . |

## Elements

### Upto conjugacy

For convenience, we take the underlying set to be .

There are seven conjugacy classes, corresponding to the unordered integer partitions of (for more information, refer cycle type determines conjugacy class). We use the notation of the cycle decomposition for permutations:

Partition | Verbal description of cycle type | Representative element with the cycle type | Size of conjugacy class | Formula calculating size | Even or odd? |
---|---|---|---|---|---|

1 + 1 + 1 + 1 + 1 | five fixed points | -- the identity element | 1 | even | |

2 + 1 + 1 + 1 | transposition: one 2-cycle, three fixed point | 10 | , also in this case | odd | |

3 + 1 + 1 | one 3-cycle, two fixed points | 20 | even | ||

2 + 2 + 1 | double transposition: two 2-cycles, one fixed point | 15 | even | ||

4 + 1 | one 4-cycle, one fixed point | 30 | odd | ||

3 + 2 | one 3-cycle, one 2-cycle | 20 | odd | ||

5 | one 5-cycle | 24 | even |

### Upto automorphism

is a complete group: in particular, every automorphism of the group is inner. Thus, the equivalence classes under automorphisms are the same as the conjugacy classes.

## Endomorphisms

### Automorphisms

Since is a complete group, it is isomorphic to its automorphism group, where each element of acts on by conjugation.

### Endomorphisms

admits three kinds of endomorphisms (that is, it admits more endomorphisms, but any endomorphism is equivalent via an automorphism to one of these three):

- The endomorphism to the trivial group
- The identity map
- The endomorphism to a group of order two, given by the sign homomorphism

## Subgroups

`Further information: Subgroup structure of symmetric group:S5`

## GAP implementation

### Group ID

This finite group has order 120 and has ID 34 among the groups of order 120 in GAP's SmallGroup library. For context, there are 47 groups of order 120. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(120,34)`

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

`gap> G := SmallGroup(120,34);`

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

`IdGroup(G) = [120,34]`

or just do:

`IdGroup(G)`

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

### Other descriptions

The group can also be defined using GAP's SymmetricGroup function as:

`SymmetricGroup(5)`