View source for Unitriangular matrix group:UT(3,p)

{{prime-parametrized particular group}}

==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===

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:

<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_{13} = a_{13} + b_{13} + a_{12}b_{23}</math>
* <math>c_{23} = a_{23} + b_{23}</math>

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}), ~~~~ (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 = e \rangle</math>

where <math>e</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>

===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==

===Automorphisms===

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|finite Heisenberg group}}