# Generating sets for subgroups of symmetric group:S3

This article gives specific information, namely, generating sets for subgroups, about a particular group, namely: symmetric group:S3.

View generating sets for subgroups of particular groups | View other specific information about symmetric group:S3

This article provides summary information on various choices of generating set for subgroups of symmetric group:S3. It builds on basic information available at element structure of symmetric group:S3 and subgroup structure of symmetric group:S3.

## Probability of generation

The rule is as follows. Given (not necessarily distinct) elements picked uniformly at random and independently of each other from a finite group , the probability that they all live in a fixed subgroup of index is .

Using this and a form of Mobius inversion on the subgroup lattice, it is possible to compute the probability that they *generate* a fixed subgroup of index (we basically need to subtract off probabilities for smaller subgroups).

### Generated by one element

Here, a single element is picked uniformly at random from the group.

Automorphism class of subgroups | List of all subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Probability that an element is in a fixed subgroup of this automorphism class(= reciprocal of index) | Probability of generating the subgroup conditional to being in the subgroup (depends only on isomorphism class) | Probability that an element generates a fixed subgroup of this automorphism class (product of preceding two columns) | Size of automorphism class | Probability that the element generates a subgroup in this automorphism class |
---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 6 | 1/6 | 1 | 1/6 | 1 | 1/6 | |

S2 in S3 | |
cyclic group:Z2 | 2 | 3 | 1/3 | 1/2 | 1/6 | 3 | 1/2 |

A3 in S3 | cyclic group:Z3 | 3 | 2 | 1/2 | 2/3 | 1/3 | 1 | 1/3 | |

whole group | |
symmetric group:S3 | 6 | 1 | 1 | 0 | 0 | 1 | 0 |

Total (4 rows) | -- | -- | -- | -- | -- | -- | -- | 6 | 1 |

### Generated by two independent possibly equal elements

Here, two elements are picked uniformly at random from the group, independent of each other. They could be equal.

Automorphism class of subgroups | List of all subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Probability that the elements are in a fixed subgroup of this automorphism class(= reciprocal square of index) | Probability of generating the subgroup conditional to being in the subgroup | Probability that the elements generates a fixed subgroup of this automorphism class (obtained by multiplying preceding two columns) | Size of automorphism class | Probability that the elements generates a subgroup in this automorphism class |
---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 6 | 1/36 | 1 | 1/36 | 1 | 1/36 | |

S2 in S3 | |
cyclic group:Z2 | 2 | 3 | 1/9 | 3/4 | 1/12 | 3 | 1/4 |

A3 in S3 | cyclic group:Z3 | 3 | 2 | 1/4 | 8/9 | 2/9 | 1 | 2/9 | |

whole group | |
symmetric group:S3 | 6 | 1 | 1 | 1/2 | 1/2 | 1 | 1/2 |

Total (4 rows) | -- | -- | -- | -- | -- | -- | -- | 6 | 1 |

### Generated by elements picked independently and uniformly at random

Below are given the expressions for general .

Automorphism class of subgroups | List of all subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Probability that the elements are in a fixed subgroup of this automorphism class(= reciprocal of index raised to power) | Probability that the elements generates a fixed subgroup of this automorphism class (obtained by Mobius inversion on preceding column) | Size of automorphism class | Probability that the elements generates a subgroup in this automorphism class |
---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 6 | 1 | ||||

S2 in S3 | |
cyclic group:Z2 | 2 | 3 | 3 | |||

A3 in S3 | cyclic group:Z3 | 3 | 2 | 1 | ||||

whole group | |
symmetric group:S3 | 6 | 1 | 1 | 1 | ||

Total (4 rows) | -- | -- | -- | -- | -- | -- | -- | 1 |

## Small generating sets for subgroups

For symmetric group:S3, the following are equivalent for any subset:

- It is a generating set of minimum size for the subgroup it generates.
- It is a minimal generating set for the subgroup it generates.

We list below all the small generating sets:

Subset | Size | Subgroup it generates | Is it a generating set of minimum size for the subgroup it generates? (if Yes, size at most 2) | Is it a minimal generating set for the subgroup it generates? (if Yes, size at most 2) | Is it a Jerrum-reduced generating set? (if Yes, size at most 2) | Is it a Sims-reduced generating set? (if Yes, size at most 3) |
---|---|---|---|---|---|---|

0 | trivial subgroup | Yes | Yes | Yes | Yes | |

1 | Yes | Yes | Yes | Yes | ||

1 | Yes | Yes | Yes | Yes | ||

1 | Yes | Yes | Yes | Yes | ||

1 | Yes | Yes | Yes | Yes | ||

1 | Yes | Yes | Yes | Yes | ||

2 | whole group | Yes | Yes | No | Yes | |

2 | whole group | Yes | Yes | Yes | Yes | |

2 | whole group | Yes | Yes | Yes | Yes | |

2 | whole group | Yes | Yes | No | No | |

2 | whole group | Yes | Yes | No | Yes | |

2 | whole group | Yes | Yes | Yes | Yes | |

2 | whole group | Yes | Yes | Yes | No | |

2 | whole group | Yes | Yes | No | No | |

2 | whole group | Yes | Yes | Yes | Yes | |

2 | A3 in S3 | No | No | No | Yes | |

3 | whole group | No | No | No | Yes | |

3 | whole group | No | No | No | Yes | |

3 | whole group | No | No | No | Yes | |

3 | whole group | No | No | No | Yes |

All other subsets fail *each* of the four questions, so they are not listed.