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

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==Definition== | ==Definition== | ||

− | + | Note that the case <math>p = 2</math>, where the group becomes [[dihedral group:D8]], behaves somewhat differently from the general case. We note on the page all the places where the discussion does not apply to <math>p = 2</math>. | |

− | + | ===As a group of matrices=== | |

− | <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 following form with the usual matrix multiplication: |

− | |||

− | |||

− | |||

− | The multiplication of matrices <math>A = (a_{ij})</math> and <math>B = (b_{ij})</math> gives the matrix < | + | <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 multiplication of matrices <math>A = (a_{ij})</math> and <math>B = (b_{ij})</math> gives the matrix <math>C = (c_{ij})</math> where: | ||

* <math>c_{12} = a_{12} + b_{12}</math> | * <math>c_{12} = a_{12} + b_{12}</math> | ||

Line 18: | Line 17: | ||

* <math>c_{23} = a_{23} + b_{23}</math> | * <math>c_{23} = a_{23} + b_{23}</math> | ||

− | The | + | The identity element is the identity matrix. |

+ | |||

+ | The inverse of a matrix <math>A = (a_{ij})</math> is the matrix <math>M = (m_{ij})</math> where: | ||

+ | |||

+ | * <math>m_{12} = -a_{12}</math> | ||

+ | * <math>m_{13} = -a_{13} + a_{12}a_{23}</math> | ||

+ | * <math>m_{23} = -a_{23}</math> | ||

+ | |||

+ | Note that all addition and multiplication in these definitions is happening over the field <math>\mathbb{F}_p</math>. | ||

+ | |||

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

+ | |||

+ | 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: | ||

+ | |||

+ | <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})</math>, | ||

+ | |||

+ | <math>(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} | ||

+ | 1 & a_{12} & a_{13}\\ | ||

+ | 0 & 1 & a_{23}\\ | ||

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

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

+ | |||

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

+ | |||

+ | 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 = 1 \rangle</math> | ||

+ | |||

+ | where <math>1</math> 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 <math>x,y,z</math> correspond to matrices: | ||

+ | |||

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

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

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

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

+ | \end{pmatrix},\ \ y=\begin{pmatrix} | ||

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

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

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

+ | \end{pmatrix},\ \ z=\begin{pmatrix} | ||

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

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

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

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

− | == | + | Note that in the above presentation, the generator <math>z</math> is redundant, and the presentation can thus be rewritten as a presentation with only two generators <math>x</math> and <math>y</math>. |

+ | |||

+ | ===As a semidirect product=== | ||

+ | |||

+ | This group of order <math>p^3</math> can also be described as a semidirect product of the [[elementary abelian group of prime-square order|elementary abelian group of order]] <math>p^2</math> by the [[group of prime order|cyclic group of order]] <math>p</math>, with the following action. Denote the base of the semidirect product as ordered pairs of elements from <math>\mathbb{Z}/p\mathbb{Z}</math>. The action of the generator of the acting group is as follows: | ||

+ | |||

+ | <math>(\alpha,\beta) \mapsto (\alpha,\alpha+\beta)</math> | ||

+ | |||

+ | In this case, for instance, we can take the subgroup with <math>a_{12} = 0</math> as the elementary abelian subgroup of order <math>p^2</math>, i.e., the elementary abelian subgroup of order <math>p^2</math> is the subgroup: | ||

+ | |||

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

+ | |||

+ | The acting subgroup of order <math>p</math> can be taken as the subgroup with <math>a_{13} = a_{23} = 0</math>, i.e., the subgroup: | ||

+ | |||

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

− | == | + | ==Families== |

− | + | # These groups fall in the more general family <math>UT(n,p)</math> of [[unitriangular matrix group]]s. The unitriangular matrix group <math>UT(n,p)</math> can be described as the group of unipotent upper-triangular matrices in <math>GL(n,p)</math>, which is also a <math>p</math>-Sylow subgroup of the [[general linear group]] <math>GL(n,p)</math>. This further can be generalized to <math>UT(n,q)</math> where <math>q</math> is the power of a prime <math>p</math>. <math>UT(n,q)</math> is the <math>p</math>-Sylow subgroup of <math>GL(n,q)</math>. | |

+ | # These groups also fall into the general family of [[extraspecial group]]s. For any number of the form <math>p^{1 + 2m}</math>, there are two extraspecial groups of that order: an extraspecial group of "+" type and an extraspecial group of "-" type. <math>UT(3,p)</math> is an extraspecial group of order <math>p^3</math> and "+" type. The other type of extraspecial group of order <math>p^3</math>, i.e., the extraspecial group of order <math>p^3</math> and "-" type, is [[semidirect product of cyclic group of prime-square order and cyclic group of prime order]]. | ||

− | + | ==Elements== | |

− | + | {{further|[[element structure of unitriangular matrix group:UT(3,p)]]}} | |

− | |||

− | === | + | ===Summary=== |

− | + | {{#lst:element structure of unitriangular matrix group:UT(3,p)|summary}} | |

− | + | ===Conjugacy class structure=== | |

− | |||

− | |||

− | + | {{#lst:element structure of unitriangular matrix group:UT(3,p)|conjugacy class summary}} | |

− | + | ==Arithmetic functions== | |

− | + | {{quotation|''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 <math>p = 2</math> and/or when <math>p = 3</math>. These are clearly indicated below. | |

− | + | ===Arithmetic functions taking values between 0 and 3=== | |

− | { | + | {| class="sortable" border="1" |

+ | ! Function !! Value !! Explanation | ||

+ | |- | ||

+ | | [[prime-base logarithm of order]] || [[arithmetic function value::prime-base logarithm of order;3|3]] || the order is <math>p^3</math> | ||

+ | |- | ||

+ | | [[prime-base logarithm of exponent]] || [[arithmetic function value::prime-base logarithm of exponent;1|1]] || the exponent is <math>p^1</math>. Exception when <math>p = 2</math>, where the exponent is <math>2^2 = 4</math>. | ||

+ | |- | ||

+ | | [[nilpotency class]] || [[arithmetic function value::nilpotency class;2|2]] || | ||

+ | |- | ||

+ | | [[derived length]] || [[arithmetic function value::derived length;2|2]] || | ||

+ | |- | ||

+ | | [[Frattini length]] || [[arithmetic function value::Frattini length;2|2]] || | ||

+ | |- | ||

+ | | [[minimum size of generating set]] || [[arithmetic function value::minimum size of generating set;2|2]] || | ||

+ | |- | ||

+ | | [[subgroup rank of a group|subgroup rank]] || [[arithmetic function value::subgroup rank of a group;2|2]] || | ||

+ | |- | ||

+ | | [[rank of a p-group|rank as p-group]] || [[arithmetic function value::rank of a p-group;2|2]] || | ||

+ | |- | ||

+ | | [[normal rank of a p-group|normal rank as p-group]] || [[arithmetic function value::normal rank of a p-group;2|2]] || | ||

+ | |- | ||

+ | | [[characteristic rank of a p-group|characteristic rank as p-group]] || [[arithmetic function value::characteristic rank of a p-group;1|1]] || | ||

+ | |} | ||

− | + | ===Arithmetic functions of a counting nature=== | |

− | + | {| class="sortable" border="1" | |

− | + | ! Function !! Value !! Explanation | |

− | + | |- | |

+ | | [[number of conjugacy classes]] || <math>\! p^2 + p - 1</math> || <math>p</math> elements in the [[center]], and each other conjugacy class has size <math>p</math> | ||

+ | |- | ||

+ | | [[number of subgroups]] || <math>\! p^2 + 2p + 4</math> when <math>p \ne 2</math>, <math>10</math> when <math>p = 2</math> || See [[subgroup structure of unitriangular matrix group:UT(3,p)]] | ||

+ | |- | ||

+ | | [[number of normal subgroups]] || <math>\! p + 4</math> || See [[subgroup structure of unitriangular matrix group:UT(3,p)]] | ||

+ | |- | ||

+ | | [[number of conjugacy classes of subgroups]] || <math>\! 2p + 5 </math> for <math>p \ne 2</math>, <math>8</math> for <math>p = 2</math> || See [[subgroup structure of unitriangular matrix group:UT(3,p)]] | ||

+ | |} | ||

− | {{ | + | ==Subgroups== |

+ | {{further|[[Subgroup structure of unitriangular matrix group:UT(3,p)]]}} | ||

− | + | Note that the analysis here specifically does ''not'' apply to the case <math>p = 2</math>. For <math>p = 2</math>, see [[subgroup structure of dihedral group:D8]]. | |

− | {{ | + | {{#lst:subgroup structure of unitriangular matrix group:UT(3,p)|summary}} |

− | + | ==Linear representation theory== | |

− | {{ | + | {{further|[[linear representation theory of unitriangular matrix group:UT(3,p)]]}} |

− | + | {{#lst:linear representation theory of unitriangular matrix group:UT(3,p)|summary}} | |

==Endomorphisms== | ==Endomorphisms== | ||

Line 79: | Line 166: | ||

The automorphisms essentially permute the subgroups of order <math>p^2</math> containing the center, while leaving the center itself unmoved. | The automorphisms essentially permute the subgroups of order <math>p^2</math> containing the center, while leaving the center itself unmoved. | ||

− | == | + | ==GAP implementation== |

+ | |||

+ | ===GAP ID=== | ||

+ | |||

+ | For any prime <math>p</math>, this group is the ''third'' group among the groups of order <math>p^3</math>. Thus, for instance, if <math>p = 7</math>, the group is described using GAP's [[GAP:SmallGroup|SmallGroup]] function as: | ||

+ | |||

+ | <tt>SmallGroup(343,3)</tt> | ||

+ | |||

+ | Note that we don't need to compute <math>p^3</math>; we can also write this as: | ||

+ | |||

+ | <tt>SmallGroup(7^3,3)</tt> | ||

+ | |||

+ | ===As an extraspecial group=== | ||

+ | |||

+ | For any prime <math>p</math>, we can define this group using GAP's [[GAP:ExtraspecialGroup|ExtraspecialGroup]] function as: | ||

+ | |||

+ | <tt>ExtraspecialGroup(p^3,'+')</tt> | ||

+ | |||

+ | For <math>p \ne 2</math>, it can also be constructed as: | ||

+ | |||

+ | <tt>ExtraspecialGroup(p^3,p)</tt> | ||

+ | |||

+ | where the argument <math>p</math> indicates that it is the extraspecial group of exponent <math>p</math>. For instance, for <math>p = 5</math>: | ||

+ | |||

+ | <tt>ExtraspecialGroup(5^3,5)</tt> | ||

− | + | ===Other descriptions=== | |

− | == | + | {| class="sortable" border="1" |

+ | ! Description !! Functions used | ||

+ | |- | ||

+ | | <tt>SylowSubgroup(GL(3,p),p)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]], [[GAP:GL|GL]] | ||

+ | |- | ||

+ | | <tt>SylowSubgroup(SL(3,p),p)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]], [[GAP:SL|SL]] | ||

+ | |- | ||

+ | | <tt>SylowSubgroup(PGL(3,p),p)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]], [[GAP:PGL|PGL]] | ||

+ | |- | ||

+ | | <tt>SylowSubgroup(PSL(3,p),p)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]], [[GAP:PSL|PSL]] | ||

+ | |} | ||

− | |||

− | |||

− | + | ==External links == | |

− | {{ | + | * {{wp|Heisenberg_group#Heisenberg_group_modulo_an_odd_prime_p}} |

## Latest revision as of 11:21, 22 August 2014

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

Note that the case , where the group becomes dihedral group:D8, behaves somewhat differently from the general case. We note on the page all the places where the discussion does not apply to .

### 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 following form with the usual matrix multiplication:

The multiplication of matrices and gives the matrix where:

The identity element is the identity matrix.

The inverse of a matrix is the matrix where:

Note that all addition and multiplication in these definitions is happening over the field .

### 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:

### 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:

Note that in the above presentation, the generator is redundant, and the presentation can thus be rewritten as a presentation with only two generators and .

### 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 , with the following action. Denote the base of the semidirect product as ordered pairs of elements from . The action of the generator of the acting group is as follows:

In this case, for instance, we can take the subgroup with as the elementary abelian subgroup of order , i.e., the elementary abelian subgroup of order is the subgroup:

The acting subgroup of order can be taken as the subgroup with , i.e., the subgroup:

## 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. For any number of the form , there are two extraspecial groups of that order: an extraspecial group of "+" type and an extraspecial group of "-" type. is an extraspecial group of order and "+" type. The other type of extraspecial group of order , i.e., the extraspecial group of order and "-" type, is semidirect product of cyclic group of prime-square order and cyclic group of prime order.

## 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)`

Note that the analysis here specifically does *not* apply to the case . For , see subgroup structure of dihedral group:D8.

### 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) |

## Endomorphisms

### Automorphisms

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

## 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)`

### Other descriptions

Description | Functions used |
---|---|

SylowSubgroup(GL(3,p),p) |
SylowSubgroup, GL |

SylowSubgroup(SL(3,p),p) |
SylowSubgroup, SL |

SylowSubgroup(PGL(3,p),p) |
SylowSubgroup, PGL |

SylowSubgroup(PSL(3,p),p) |
SylowSubgroup, PSL |