# Presentations for groups of prime-cube order

This article gives specific information, namely, presentations, about a family of groups, namely: groups of prime-cube order.

View presentations for group families | View other specific information about groups of prime-cube order

The discussion here applies in full to . The behavior for is somewhat different. See presentations for groups of order 8 for more details.

## Power-commutator presentations

Each of the power-commutator presentations uses three generators . The power relations are of the form where are natural numbers that depend on the nature of the group, and where depends on the nature of the group. The commutator relation is of the form where depends on the nature of the group. Note that , , and are always the identity.

It turns out that for the isomorphism class of the final group, all the four values matter only mod . Thus, for simplicity, we assume that they are in the set . Note that this is a general feature of power-commutator presentations.

### Simplified power-commutator presentations

We here provide a *single* power-commutator presentation among the many possibilities.

Group | Second part of GAP ID (GAP ID is (p^3,2nd part) | Nilpotency class | Minimum size of generating set | Prime-base logarithm of exponent | full power-commutator presentation | ||||
---|---|---|---|---|---|---|---|---|---|

cyclic group of prime-cube order | 1 | 1 | 1 | 3 | 1 | 0 | 1 | 0 | [SHOW MORE] |

direct product of cyclic group of prime-square order and cyclic group of prime order | 2 | 1 | 2 | 2 | 0 | 1 | 0 | 0 | [SHOW MORE] |

prime-cube order group:U(3,p) | 3 | 2 | 2 | 1 | 0 | 0 | 0 | 1 | [SHOW MORE] |

semidirect product of cyclic group of prime-square order and cyclic group of prime order | 4 | 2 | 2 | 2 | 0 | 1 | 0 | 1 | [SHOW MORE] |

elementary abelian group of prime-cube order | 5 | 1 | 3 | 1 | 0 | 0 | 0 | 0 | [SHOW MORE] |

### Determining the isomorphism class from an arbitrary power-commutator presentation

Note that for the second and fourth groups, there are multiple sets of possible conditions, given in two separate rows within those groups. These are equivalent under permutations of the generators

Group | Second part of GAP ID (GAP ID is (p^3,2nd part) | Nilpotency class | Minimum size of generating set | Condition on | Condition on | Condition on | Condition on |
---|---|---|---|---|---|---|---|

cyclic group of prime-cube order | 1 | 1 | 1 | nonzero | arbitrary | nonzero | zero |

direct product of cyclic group of prime-square order and cyclic group of prime order | 2 | 1 | 2 | arbitrary nonzero zero nonzero |
nonzero arbitrary zero zero |
zero zero nonzero nonzero |
zero zero zero zero |

prime-cube order group:U(3,p) | 3 | 2 | 3 | zero | zero | zero | nonzero |

semidirect product of cyclic group of prime-square order and cyclic group of prime order | 4 | 2 | 2 | zero zero |
zero nonzero |
nonzero zero |
nonzero nonzero |

elementary abelian group of prime-cube order | 5 | 1 | 3 | zero | zero | zero | zero |

The following algorithmic approach is a little faster:

Condition letter | Condition detail | If yes, then possible GAP IDs | If no, then possible GAP IDs |
---|---|---|---|

A | 1,2,5 (abelian groups) | 3,4 (non-abelian groups) | |

B | All: |
3,5 (exponent groups) | 1,2,4 |

C | All: , |
1,4 | 2,3,5 |

We see now that each ID has a unique condition letter combination:

Group | Second part of GAP ID (GAP ID is (p^3,2nd part) | Nilpotency class | Minimum size of generating set | Condition letter combination |
---|---|---|---|---|

cyclic group of prime-cube order | 1 | 1 | 1 | satisfies A and C but not B |

direct product of cyclic group of prime-square order and cyclic group of prime order | 2 | 1 | 2 | satisfies A but not B or C |

prime-cube order group:U(3,p) | 3 | 2 | 3 | satisfies B but not A or C |

semidirect product of cyclic group of prime-square order and cyclic group of prime order | 4 | 2 | 2 | satisfies C but not A or B |

elementary abelian group of prime-cube order | 5 | 1 | 3 | satisfies A and B but not C (in fact, it suffices to check A and B and ignore C) |