# Element structure of groups of order 81

## Contents

## 1-isomorphism

### Pairs where one of the groups is abelian

Of the 15 groups of order 81, 5 are abelian, 6 have nilpotency class two, and 4 have nilpotency class three. Via the Baer correspondence, each of the groups of class two has a Baer Lie ring, and in particular is 1-isomorphic to the additive group of that Lie ring. Of the 4 groups of nilpotency class three, only one (SmallGroup(81,8)) is 1-isomorphic to an abelian group. There are *no 1-isomorphisms between pairs where both members are non-abelian.*

### Grouping by abelian member

Abelian member | GAP ID second part | Total number of members (including abelian member) | Other members | GAP IDs second part (in order of listing) |
---|---|---|---|---|

direct product of Z9 and Z9 | 2 | 2 | nontrivial semidirect product of Z9 and Z9 | 4 |

direct product of Z27 and Z3 | 5 | 2 | M81 | 6 |

direct product of Z9 and E9 | 11 | 5 | SmallGroup(81,3), SmallGroup(81,8), direct product of semidirect product of Z9 and Z3 and Z3, central product of prime-cube order group:U(3,3) and Z9 | 3, 8, 13, 14 |

elementary abelian group:E81 | 15 | 2 | direct product of prime-cube order group:U(3,3) and Z3 | 12 |

### Groupings that do not have any abelian members

There are *no 1-isomorphisms between pairs where both members are non-abelian* for order 81. Thus, there are no such groupings.

## Order statistics

FACTS TO CHECK AGAINST:

ORDER STATISTICS(cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots

1-ISOMORPHISM(cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic

### Order statistics raw data

Group | Second part of GAP ID | Number of elements of order 1 | Number of elements of order 3 | Number of elements of order 9 | Number of elements of order 27 | Number of elements of order 81 |
---|---|---|---|---|---|---|

Cyclic group:Z81 | 1 | 1 | 2 | 6 | 18 | 54 |

Direct product of Z9 and Z9 | 2 | 1 | 8 | 72 | 0 | 0 |

SmallGroup(81,3) | 3 | 1 | 26 | 54 | 0 | 0 |

Nontrivial semidirect product of Z9 and Z9 | 4 | 1 | 8 | 72 | 0 | 0 |

Direct product of Z27 and Z3 | 5 | 1 | 8 | 18 | 54 | 0 |

M81 | 6 | 1 | 8 | 18 | 54 | 0 |

Wreath product of Z3 and Z3 | 7 | 1 | 44 | 36 | 0 | 0 |

SmallGroup(81,8) | 8 | 1 | 26 | 54 | 0 | 0 |

SmallGroup(81,9) | 9 | 1 | 62 | 18 | 0 | 0 |

SmallGroup(81,10) | 10 | 1 | 8 | 72 | 0 | 0 |

Direct product of Z9 and E9 | 11 | 1 | 26 | 54 | 0 | 0 |

Direct product of prime-cube order group:U(3,3) and Z3 | 12 | 1 | 80 | 0 | 0 | 0 |

Direct product of semidirect product of Z9 and Z3 and Z3 | 13 | 1 | 26 | 54 | 0 | 0 |

SmallGroup(81,14) | 14 | 1 | 26 | 54 | 0 | 0 |

Elementary abelian group:E81 | 15 | 1 | 80 | 0 | 0 | 0 |

Here are the cumulative order statistics.

Group | Second part of GAP ID | Number of 1st roots | Number of 3rd roots | Number of 9th roots | Number of 27th roots | Number of 81th roots |
---|---|---|---|---|---|---|

Cyclic group:Z81 | 1 | 1 | 3 | 9 | 27 | 81 |

Direct product of Z9 and Z9 | 2 | 1 | 9 | 81 | 81 | 81 |

SmallGroup(81,3) | 3 | 1 | 27 | 81 | 81 | 81 |

Nontrivial semidirect product of Z9 and Z9 | 4 | 1 | 9 | 81 | 81 | 81 |

Direct product of Z27 and Z3 | 5 | 1 | 9 | 27 | 81 | 81 |

M81 | 6 | 1 | 9 | 27 | 81 | 81 |

Wreath product of Z3 and Z3 | 7 | 1 | 45 | 81 | 81 | 81 |

SmallGroup(81,8) | 8 | 1 | 27 | 81 | 81 | 81 |

SmallGroup(81,9) | 9 | 1 | 63 | 81 | 81 | 81 |

SmallGroup(81,10) | 10 | 1 | 9 | 81 | 81 | 81 |

Direct product of Z9 and E9 | 11 | 1 | 27 | 81 | 81 | 81 |

Direct product of prime-cube order group:U(3,3) and Z3 | 12 | 1 | 81 | 81 | 81 | 81 |

Direct product of semidirect product of Z9 and Z3 and Z3 | 13 | 1 | 27 | 81 | 81 | 81 |

SmallGroup(81,14) | 14 | 1 | 27 | 81 | 81 | 81 |

Elementary abelian group:E81 | 15 | 1 | 81 | 81 | 81 | 81 |

### Equivalence classes based on order statistics

Here, we discuss the equivalence classes of groups of order 81 up to being order statistics-equivalent finite groups and up to the stronger notion of being 1-isomorphic groups (which means there is a bijection that restricts to isomorphisms on cyclic subgroups). See also order statistics-equivalent not implies 1-isomorphic.

Order statistics | Order statistics (cumulative) | Number of groups | Number of equivalence classes up to 1-isomorphism | Members of first equivalence class | Members of second equivalence class | Abelian group with these order statistics? | Cumulative order statistics all powers of 3? |
---|---|---|---|---|---|---|---|

1,2,6,18,54 | 1,3,9,27,81 | 1 | 1 | Cyclic group:Z81 (ID:1) | -- | Yes | Yes |

1,8,18,54,0 | 1,9,27,81,81 | 2 | 1 | Direct product of Z27 and Z3 (ID:5) and M81 (ID:6) | -- | Yes | Yes |

1,8,72,0,0 | 1,9,81,81,81 | 3 | 2 | Direct product of Z9 and Z9 (ID:2), Semidirect product of Z9 and Z9 (ID:4) | SmallGroup(81,10) (ID:10) | Yes | Yes |

1,26,54,0,0 | 1,27,81,81,81 | 5 | 1 or 2 | Direct product of Z9 and E9 (ID:11), SmallGroup(81,3) (ID:3), Direct product of semidirect product of Z9 and Z3 and Z3 (ID:13), Central product of prime-cube order group:U(3,3) and Z9 (ID:14), SmallGroup(81,8) (ID:8) | Yes | Yes | |

1,44,36,0,0 | 1,45,81,81,81 | 1 | 1 | Wreath product of Z3 and Z3 (ID:7) | -- | No | No |

1,62,18,0,0 | 1,63,81,81,81 | 1 | 1 | SmallGroup(81,9) (ID:9) | -- | No | No |

1,80,0,0,0 | 1,81,81,81,81 | 2 | 1 | Elementary abelian group:E81 (ID:15) and Direct product of prime-cube order group:U(3,3) and Z3 (ID:12) | -- | Yes | Yes |