# Unitriangular matrix group:UT(3,p)

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.

View other such prime-parametrized groups

## Contents

## Definition

### As a group of matrices

Given a prime , the group is defined as the unitriangular matrix group of degree three over the prime field .

The analysis given below does not apply to the case . For , we get the dihedral group:D8, which is studied separately.

### As a semidirect product

This group of order can also be described as a semidirect product of the elementary abelian group of order by the cyclic group of order , where the generator of the cyclic group of order acts via the automorphism:

In this case, for instance, we can take the subgroup with as the elementary abelian subgroup of order and the subgroup with as the cyclic subgroup of order .

## Families

- These groups fall in the more general family of unipotent upper-triangular matrices, which is the -Sylow subgroup of the general linear group . This further can be generalized to where is the power of a prime , which is the -Sylow subgroup of .
- These groups also fall into the general family of extraspecial groups.

## Elements

### Upto conjugacy

Every element has order .

The conjugacy classes are as follows:

- The center has order precisely , so there are elements that form conjugacy classes of size 1. These are, specifically, the elements with , and they're thus parametrized by their entry.
- For every element outside the center, the centralizer of that element is the subgroup generated by that element and the center, and is hence of order . Thus, the conjugacy class of the element is of size . Thus, there is a total of conjugacy classes of size .

### Upto automorphism

There are only three classes of elements upto automorphism:

- The identity element, which forms a class of size 1
- The non-identity elements in the center, which form a class of size
- The non-central elements, which form a class of size

## Arithmetic functions

Compare and contrast arithmetic function values with other groups of prime-cube order at Groups of prime-cube order#Arithmetic functions

For some of these, the function values are different when and/or when . These are clearly indicated below.

### Arithmetic functions taking values between 0 and 3

Function | Value | Explanation |
---|---|---|

prime-base logarithm of order | 3 | the order is |

prime-base logarithm of exponent | 1 | the exponent is . Exception when , where the exponent is . |

nilpotency class | 2 | |

derived length | 2 | |

Frattini length | 2 | |

minimum size of generating set | 2 | |

subgroup rank | 2 | |

rank as p-group | 2 | |

normal rank as p-group | 2 | |

characteristic rank as p-group | 1 |

### Arithmetic functions of a counting nature

Function | Value | Explanation |
---|---|---|

number of conjugacy classes | elements in the center, and each other conjugacy class has size | |

number of subgroups | when , when | |

number of normal subgroups | ||

number of conjugacy classes of subgroups | for , for |

## Subgroups

`Further information: Subgroup structure of prime-cube order group:U(3,p)`
Here is the complete list of subgroups:

- The trivial subgroup (1)
- The center, which is a group of order . In matrix terms, this is the subgroup comprising matrices with . (1)
- Subgroups of order generated by non-central elements. These are not normal, and occur in conjugacy classes of size . ()
- Subgroups of order containing the center. These are the inverse images via the quotient map by the center, of subgroups of order in the inner automorphism group. ()
- The whole group. (1)

### Normal subgroups

The subgroups in (1), (2), (4) and (5) above are normal.

### Characteristic subgroups

The subgroups in (1), (2) and (5) above are normal. In other words, there are only three characteristic subgroups. Some notable facts:

- The group is characteristic-comparable: any two characteristic subgroups can be compared
- More generally, any characteristic subgroup and any normal subgroup can be compared.
- The characteristic subgroups are precisely the subgroups that occur in the derived series, upper central series and lower central series.

## Subgroup-defining functions

Subgroup-defining function | Subgroup type in list | Isomorphism class | Comment |
---|---|---|---|

Center | (2) | Group of prime order | |

Commutator subgroup | (2) | Group of prime order | |

Frattini subgroup | (2) | Group of prime order | The maximal subgroups of order intersect here. |

Socle | (2) | Group of prime order | This subgroup is the unique minimal normal subgroup, i.e.,the monolith, and the group is monolithic. Also, minimal normal implies central in nilpotent. |

### Quotient-defining function

Quotient-defining function | Isomorphism class | Comment |
---|---|---|

Inner automorphism group | Elementary abelian group of prime-square order | It is the quotient by the center, which is of prime order. |

Abelianization | Elementary abelian group of prime-square order | It is the quotient by the commutator subgroup, which is of prime order. |

Frattini quotient | Elementary abelian group of prime-square order | It is the quotient by the Frattini subgroup, which is of prime order. |

## GAP implementation

### GAP ID

For any prime , this group is the *third* group among the groups of order . Thus, for instance, if , the group is described using GAP's SmallGroup function as:

`SmallGroup(343,3)`

Note that we don't need to compute ; we can also write this as:

`SmallGroup(7^3,3)`

### As an extraspecial group

For any prime , we can define this group using GAP's ExtraspecialGroup function as:

`ExtraspecialGroup(p^3,'+')`

For , it can also be constructed as:

`ExtraspecialGroup(p^3,p)`

where the argument indicates that it is the extraspecial group of exponent . For instance, for :

`ExtraspecialGroup(5^3,5)`

## Endomorphisms

### Automorphisms

The automorphisms essentially permute the subgroups of order containing the center, while leaving the center itself unmoved.

## Related groups

For any prime , there are (up to isomorphism) two non-abelian groups of order . One of them is this, and the other is the semidirect product of the cyclic group of order by a group of order acting by power maps (with the generator corresponding to multiplication by ).