# Symmetric group:S5

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## 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., .
- It is the projective semilinear group of degree two over the field of four elements, i.e., .

### Presentation

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 120#Arithmetic functions

## 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

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

### 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 | Partition in grouped form | Verbal description of cycle type | Representative element with the cycle type | Size of conjugacy class | Formula calculating size | Even or odd? If even, splits? If splits, real in alternating group? | Element order | Formula calcuating element order |
---|---|---|---|---|---|---|---|---|

1 + 1 + 1 + 1 + 1 | 1 (5 times) | five fixed points | -- the identity element | 1 | even; no | 1 | ||

2 + 1 + 1 + 1 | 2 (1 time), 1 (3 times) | transposition: one 2-cycle, three fixed point | 10 | or , also in this case | odd | 2 | ||

3 + 1 + 1 | 3 (1 time), 1 (2 times) | one 3-cycle, two fixed points | 20 | or | even; no | 3 | ||

2 + 2 + 1 | 2 (2 times), 1 (1 time) | double transposition: two 2-cycles, one fixed point | 15 | or | even; no | 2 | ||

4 + 1 | 4 (1 time), 1 (1 time) | one 4-cycle, one fixed point | 30 | or | odd | 4 | ||

3 + 2 | 3 (1 time), 2 (1 time) | one 3-cycle, one 2-cycle | 20 | or | odd | 6 | ||

5 | 5 (1 time) | one 5-cycle | 24 | or | even; yes; yes | 5 | ||

Total (7 rows, 7 being the number of unordered integer partitions of 5) | -- | -- | -- | 120 (equals order of the group) | -- | odd: 60 (3 classes) even;no: 36 (3 classes) even;yes;yes: 24 (1 class) |
[SHOW MORE] |

### 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. In fact, is complete for . See symmetric groups on finite sets are complete.

## Endomorphisms

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

### Summary information

Construct | Value | Order | Second part of GAP ID (if group) |
---|---|---|---|

endomorphism monoid | ? | 146 | -- |

automorphism group | symmetric group:S5 | 120 | 34 |

inner automorphism group | symmetric group:S5 | 120 | 34 |

extended automorphism group | direct product of S5 and Z2 | 240 | 189 |

### Automorphisms

Since is a complete group, it is isomorphic to its automorphism group, where each element of acts on by conjugation. `Further information: symmetric groups on finite sets are complete`

## 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

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

SymmetricGroup(5) |
SymmetricGroup |

PGL(2,5) |
PGL |