# Difference between revisions of "Unitriangular matrix group:UT(3,p)"

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===As a group of matrices=== | ===As a group of matrices=== | ||

− | Given a prime <math>p</math>, the group <math>UT(3,p)</math> is defined as the [[unitriangular matrix group]] of [[unitriangular matrix group of degree three|degree three]] over the [[prime field]] <math>\mathbb{F}_p</math>. | + | Given a prime <math>p</math>, the group <math>UT(3,p)</math> is defined as the [[unitriangular matrix group]] of [[unitriangular matrix group of degree three|degree three]] over the [[prime field]] <math>\mathbb{F}_p</math>. Explicitly, it has the form: |

+ | |||

+ | <math>\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{23} \in \mathbb{F}_p \right \}</math> | ||

The analysis given below does not apply to the case <math>p = 2</math>. For <math>p = 2</math>, we get the [[dihedral group:D8]], which is studied separately. | The analysis given below does not apply to the case <math>p = 2</math>. For <math>p = 2</math>, we get the [[dihedral group:D8]], which is studied separately. | ||

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===Definition by presentation=== | ===Definition by presentation=== | ||

− | The group can be defined by | + | The group can be defined by means of the following [[presentation]]: |

− | + | <math>\langle x,y,z \mid [x,y] = z, xz = zx, yz = zy, x^p = y^p = z^p = e \rangle</math> | |

− | These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics | + | where <math>e</math> denotes the identity element. |

− | and so the group is sometimes called | + | |

+ | These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics and so the group is sometimes called a finite Heisenberg group. Generators <math>x,y,z</math> correspond to matrices: | ||

:<math>x=\begin{pmatrix} | :<math>x=\begin{pmatrix} | ||

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===In coordinate form=== | ===In coordinate form=== | ||

− | We may define the group as set of triples | + | We may define the group as set of triples <math>(a_{12},a_{13},a_{23})</math> over the [[prime field]] <math>\mathbb{F}_p</math>, |

with the multiplication law given by: | with the multiplication law given by: | ||

− | :<math> ( | + | :<math> (a_{12},a_{13},a_{23}) (b_{12},b_{13},b_{23}) = (a_{12} + b_{12},a_{13} + b_{13} + a_{12}b_{23}, a_{23} + b_{23}), ~~~~ (a_{12},a_{13},a_{23})^{-1} = (-a_{12}, -a_{13} + a_{12}a_{23}, a_{23}) </math>. |

+ | |||

+ | The matrix corresponding to triple <math>(a_{12},a_{13},a_{23})</math> is: | ||

− | |||

:<math>\begin{pmatrix} | :<math>\begin{pmatrix} | ||

− | 1 & | + | 1 & a_{12} & a_{13}\\ |

− | 0 & 1 & | + | 0 & 1 & a_{23}\\ |

0 & 0 & 1\\ | 0 & 0 & 1\\ | ||

\end{pmatrix}</math> | \end{pmatrix}</math> |

## Revision as of 14:45, 18 September 2012

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 . Explicitly, it has the form:

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 .

### Definition by presentation

The group can be defined by means of the following presentation:

where denotes the identity element.

These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics and so the group is sometimes called a finite Heisenberg group. Generators correspond to matrices:

### In coordinate form

We may define the group as set of triples over the prime field , with the multiplication law given by:

- .

The matrix corresponding to triple is:

## Families

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

## Elements

`Further information: element structure of unitriangular matrix group:UT(3,p)`

### Summary

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

number of conjugacy classes | |

order | Agrees with general order formula for : |

conjugacy class size statistics | size 1 ( times), size ( times) |

orbits under automorphism group | Case : size 1 (1 conjugacy class of size 1), size 1 (1 conjugacy class of size 1), size 2 (1 conjugacy class of size 2), size 4 (2 conjugacy classes of size 2 each) Case odd : size 1 (1 conjugacy class of size 1), size ( conjugacy classes of size 1 each), size ( conjugacy classes of size each) |

number of orbits under automorphism group | 4 if 3 if is odd |

order statistics | Case : order 1 (1 element), order 2 (5 elements), order 4 (2 elements) Case odd: order 1 (1 element), order ( elements) |

exponent | 4 if if odd |

### Conjugacy class structure

Note that the characteristic polynomial of all elements in this group is , hence we do not devote a column to the characteristic polynomial.

For reference, we consider matrices of the form:

Nature of conjugacy class | Jordan block size decomposition | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Order of elements in each such conjugacy class | Type of matrix |
---|---|---|---|---|---|---|---|

identity element | 1 + 1 + 1 + 1 | 1 | 1 | 1 | 1 | ||

non-identity element, but central (has Jordan blocks of size one and two respectively) | 2 + 1 | 1 | , | ||||

non-central, has Jordan blocks of size one and two respectively | 2 + 1 | , but not both and are zero | |||||

non-central, has Jordan block of size three | 3 | if odd 4 if |
both and are nonzero | ||||

Total (--) | -- | -- | -- | -- | -- |

## 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 | See subgroup structure of unitriangular matrix group:UT(3,p) |

number of normal subgroups | See subgroup structure of unitriangular matrix group:UT(3,p) | |

number of conjugacy classes of subgroups | for , for | See subgroup structure of unitriangular matrix group:UT(3,p) |

## Subgroups

`Further information: Subgroup structure of unitriangular matrix group:UT(3,p)`

### Table classifying subgroups up to automorphisms

Automorphism class of subgroups | Representative | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes | Size of each conjugacy class | Number of subgroups | Isomorphism class of quotient (if exists) | Subnormal depth (if subnormal) |
---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 1 | 1 | 1 | prime-cube order group:U(3,p) | 1 | ||

center of unitriangular matrix group:UT(3,p) | ; equivalently, given by . | group of prime order | 1 | 1 | 1 | elementary abelian group of prime-square order | 1 | ||

non-central subgroups of prime order in unitriangular matrix group:UT(3,p) | Subgroup generated by any element with at least one of the entries nonzero | group of prime order | -- | 2 | |||||

elementary abelian subgroups of prime-square order in unitriangular matrix group:UT(3,p) | join of center and any non-central subgroup of prime order | elementary abelian group of prime-square order | 1 | group of prime order | 1 | ||||

whole group | all elements | unitriangular matrix group:UT(3,p) | 1 | 1 | 1 | 1 | trivial group | 0 | |

Total (5 rows) | -- | -- | -- | -- | -- | -- | -- |

### Tables classifying isomorphism types of subgroups

Group name | GAP ID | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|

Trivial group | 1 | 1 | 1 | 1 | |

Group of prime order | 1 | 1 | |||

Elementary abelian group of prime-square order | 0 | ||||

Prime-cube order group:U3p | 1 | 1 | 1 | 1 | |

Total | -- |

### Table listing number of subgroups by order

Group order | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|

1 | 1 | 1 | 1 | |

1 | 1 | |||

0 | ||||

1 | 1 | 1 | 1 | |

Total |

## Linear representation theory

`Further information: linear representation theory of unitriangular matrix group:UT(3,p)`

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

number of conjugacy classes (equals number of irreducible representations over a splitting field) | . See number of irreducible representations equals number of conjugacy classes, element structure of unitriangular matrix group of degree three over a finite field |

degrees of irreducible representations over a splitting field (such as or ) | 1 (occurs times), (occurs times) |

sum of squares of degrees of irreducible representations | (equals order of the group) see sum of squares of degrees of irreducible representations equals order of group |

lcm of degrees of irreducible representations | |

condition for a field (characteristic not equal to ) to be a splitting field | The polynomial should split completely. For a finite field of size , this is equivalent to . |

field generated by character values, which in this case also coincides with the unique minimal splitting field (characteristic zero) | Field where is a primitive root of unity. This is a degree extension of the rationals. |

unique minimal splitting field (characteristic ) | The field of size where is the order of mod . |

degrees of irreducible representations over the rational numbers | 1 (1 time), ( times), (1 time) |

Orbits over a splitting field under the action of the automorphism group | Case : Orbit sizes: 1 (degree 1 representation), 1 (degree 1 representation), 2 (degree 1 representations), 1 (degree 2 representation) Case odd : Orbit sizes: 1 (degree 1 representation), (degree 1 representations), (degree representations) number: 4 (for ), 3 (for odd ) |

Orbits over a splitting field under the multiplicative action of one-dimensional representations | Orbit sizes: (degree 1 representations), and orbits of size 1 (degree representations) |

## 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 ).