# Fusion systems for quaternion group

From Groupprops

This article gives specific information, namely, fusion systems, about a particular group, namely: quaternion group.

View fusion systems for particular groups | View other specific information about quaternion group

This article discusses possible fusion systems for the quaternion group.

## Summary

Item | Value |
---|---|

Total number of saturated fusion systems on a concrete instance of the group (strict, not up to isomorphism of fusion systems) |
2 |

Total number of saturated fusion systems up to isomorphism | 2 |

List of saturated fusion systems with orbit sizes | inner fusion system (orbit size 1 under isomorphisms), non-inner fusion system for quaternion group (orbit size 1 under isomorphisms) |

Number of simple fusion systems | None (the reason is that in every fusion system, the fusion subsystem induced on the center of quaternion group is normal) |

Number of maximal fusion systems, i.e., fusion systems not contained in bigger fusion systems | 1 (the non-inner case) |

## Description of saturated fusion systems

Isomorphism type of fusion system | Number of such fusion systems under strict counting | Can the fusion system be realized using a Sylow subgroup of a finite group? | Does the identity functor control strong fusion? This would mean that all fusion occurs in the normalizer | Is the fusion system simple? | Smallest size embedding realizing this fusion system (if any) |
---|---|---|---|---|---|

inner fusion system | 1 | Yes | Yes | No | as a subgroup of itself |

non-inner fusion system for quaternion group | 1 | Yes | Yes | No | Q8 in SL(2,3) |

## Inner fusion system

Equivalence class under isomorphisms | Subgroups involved | Order | Index | Total number of subgroups | Are all group automorphisms of each subgroup included? | Size of automorphism group from the fusion system | Total number of isomorphisms (including automorphisms and others) = (number of automorphisms) (number of subgroups) |
---|---|---|---|---|---|---|---|

trivial subgroup | trivial subgroup | 1 | 8 | 1 | Yes | 1 | 1 |

center of quaternion group | 2 | 4 | 1 | Yes | 1 | 1 | |

cyclic maximal subgroups of quaternion group | 4 | 2 | 1 | Yes | 2 | 2 | |

cyclic maximal subgroups of quaternion group | 4 | 2 | 1 | Yes | 2 | 2 | |

cyclic maximal subgroups of quaternion group | 4 | 2 | 1 | Yes | 2 | 2 | |

whole group | whole group | 8 | 1 | 1 | No | 4 | 4 |

### Sylow subgroups realizing this fusion system

This fusion system is realized by a group having quaternion group as its 2-Sylow subgroup if and if it possesses a normal complement, so the 2-Sylow subgroup is a retract of the group and the group is a semidirect product of a normal -subgroup and the quaternion group, or equivalently the group is a 2-nilpotent group.

Some examples are below:

Group | Order | Isomorphism class of normal complement | Is it a direct product? |
---|---|---|---|

direct product of Q8 and Z3 | 24 | cyclic group:Z3 | Yes |

dicyclic group:Dic24 | 24 | cyclic group:Z3 | No |

direct product of Q8 and Z5 | 40 | cyclic group:Z5 | Yes |

dicyclic group:Dic40 | 40 | cyclic group:Z5 | No |

## Fusion system using an outer automorphism of order three

Equivalence class under isomorphisms | Subgroups involved | Order | Index | Number of subgroups (=1 iff weakly closed subgroup) | Are all group automorphisms of each subgroup included? | Size of automorphism group from the fusion system | Total number of isomorphisms (including automorphisms and others) = (number of automorphisms) (number of subgroups) |
---|---|---|---|---|---|---|---|

trivial subgroup | trivial subgroup | 1 | 8 | 1 | Yes | 1 | 1 |

center of quaternion group | 2 | 4 | 1 | Yes | 1 | 1 | |

, , | cyclic maximal subgroups of quaternion group | 4 | 2 | 3 | Yes | 2 | 18 |

whole group | whole group | 8 | 1 | 1 | Yes | 24 | 24 |

### Sylow subgroups realizing this fusion system

Any situation where quaternion group arises as a 2-Sylow subgroup that is *not* a retract, i.e., does *not* have a normal complement. Any such example must admit special linear group:SL(2,3) as a subquotient. Examples are given below:

Group | Order | Quaternion group as a subgroup of this group | Comment |
---|---|---|---|

special linear group:SL(2,3) | 24 | Q8 in SL(2,3) | |

special linear group:SL(2,5) | 120 | Q8 in SL(2,5) | There is an intermediate SL(2,3) in SL(2,5) that controls all the fusion behavior. |