# Generating sets for subgroups of quaternion group

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 quaternion group and subgroup structure of quaternion group.

## 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 that an element generates a fixed subgroup of this automorphism class (obtained by Mobius inversion on preceding column) | Size of automorphism class | Probability that the element generates a subgroup in this automorphism class |
---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial subgroup | 1 | 8 | 1/8 | 1/8 | 1 | 1/8 | |

center of quaternion group | cyclic group:Z2 | 2 | 4 | 1/4 | 1/8 | 1 | 1/8 | |

cyclic maximal subgroups of quaternion group | |
cyclic group:Z4 | 4 | 2 | 1/2 | 1/4 | 3 | 3/4 |

whole group | quaternion group | 8 | 1 | 1 | 0 | 1 | 0 | |

Total | -- | -- | -- | -- | -- | -- | -- | 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 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 subgroup | 1 | 8 | 1/64 | 1/64 | 1 | 1/64 | |

center of quaternion group | cyclic group:Z2 | 2 | 4 | 1/16 | 3/64 | 1 | 3/64 | |

cyclic maximal subgroups of quaternion group | |
cyclic group:Z4 | 4 | 2 | 1/4 | 3/16 | 3 | 9/16 |

whole group | quaternion group | 8 | 1 | 1 | 3/8 | 1 | 3/8 | |

Total | -- | -- | -- | -- | -- | -- | -- | 1 |