Unitriangular matrix group:UT(3,p)

Definition
Note that the case $$p = 2$$, 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 $$p = 2$$.

As a group of matrices
Given a prime $$p$$, the group $$UT(3,p)$$ is defined as the unitriangular matrix group of degree three over the prime field $$\mathbb{F}_p$$. Explicitly, it has the following form with the usual matrix multiplication:

$$\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{23} \in \mathbb{F}_p \right \}$$

The multiplication of matrices $$A = (a_{ij})$$ and $$B = (b_{ij})$$ gives the matrix $$C = (c_{ij})$$ where:


 * $$c_{12} = a_{12} + b_{12}$$
 * $$c_{13} = a_{13} + b_{13} + a_{12}b_{23}$$
 * $$c_{23} = a_{23} + b_{23}$$

The identity element is the identity matrix.

The inverse of a matrix $$A = (a_{ij})$$ is the matrix $$M = (m_{ij})$$ where:


 * $$m_{12} = -a_{12}$$
 * $$m_{13} = -a_{13} + a_{12}a_{23}$$
 * $$m_{23} = -a_{23}$$

Note that all addition and multiplication in these definitions is happening over the field $$\mathbb{F}_p$$.

In coordinate form
We may define the group as set of triples $$(a_{12},a_{13},a_{23})$$ over the prime field $$\mathbb{F}_p$$, with the multiplication law given by:

$$ (a_{12},a_{13},a_{23}) (b_{12},b_{13},b_{23}) = (a_{12} + b_{12},a_{13} + b_{13} + a_{12}b_{23}, a_{23} + b_{23}), ~ (a_{12},a_{13},a_{23})^{-1} = (-a_{12}, -a_{13} + a_{12}a_{23}, -a_{23}) $$.

The matrix corresponding to triple $$(a_{12},a_{13},a_{23})$$ is:


 * $$\begin{pmatrix}

1 & a_{12} & a_{13}\\ 0 & 1 & a_{23}\\ 0 & 0 & 1\\ \end{pmatrix}$$

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

$$\langle x,y,z \mid [x,y] = z, xz = zx, yz = zy, x^p = y^p = z^p = 1 \rangle$$

where $$1$$ 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 $$x,y,z$$ correspond to matrices:


 * $$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}$$

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

As a semidirect product
This group of order $$p^3$$ can also be described as a semidirect product of the elementary abelian group of order $$p^2$$ by the cyclic group of order $$p$$, with the following action. Denote the base of the semidirect product as ordered pairs of elements from $$\mathbb{Z}/p\mathbb{Z}$$. The action of the generator of the acting group is as follows:

$$(\alpha,\beta) \mapsto (\alpha,\alpha+\beta)$$

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

$$\left \{ \begin{pmatrix} 1 & 0 & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{13}, a_{23} \in \mathbb{F}_p \right \}$$

The acting subgroup of order $$p$$ can be taken as the subgroup with $$a_{13} = a_{23} = 0$$, i.e., the subgroup:

$$\left \{ \begin{pmatrix} 1 & a_{12} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12} \in \mathbb{F}_p \right \}$$

Families

 * 1) These groups fall in the more general family $$UT(n,p)$$ of unitriangular matrix groups. The unitriangular matrix group $$UT(n,p)$$ can be described as the group of unipotent upper-triangular matrices in $$GL(n,p)$$, which is also a $$p$$-Sylow subgroup of the general linear group $$GL(n,p)$$. This further can be generalized to $$UT(n,q)$$ where $$q$$ is the power of a prime $$p$$. $$UT(n,q)$$ is the $$p$$-Sylow subgroup of $$GL(n,q)$$.
 * 2) These groups also fall into the general family of extraspecial groups. For any number of the form $$p^{1 + 2m}$$, there are two extraspecial groups of that order: an extraspecial group of "+" type and an extraspecial group of "-" type. $$UT(3,p)$$ is an extraspecial group of order $$p^3$$ and "+" type. The other type of extraspecial group of order $$p^3$$, i.e., the extraspecial group of order $$p^3$$ and "-" type, is semidirect product of cyclic group of prime-square order and cyclic group of prime order.

Arithmetic functions
For some of these, the function values are different when $$p = 2$$ and/or when $$p = 3$$. These are clearly indicated below.

Subgroups
Note that the analysis here specifically does not apply to the case $$p = 2$$. For $$p = 2$$, see subgroup structure of dihedral group:D8.

Automorphisms
The automorphisms essentially permute the subgroups of order $$p^2$$ containing the center, while leaving the center itself unmoved.

GAP ID
For any prime $$p$$, this group is the third group among the groups of order $$p^3$$. Thus, for instance, if $$p = 7$$, the group is described using GAP's SmallGroup function as:

SmallGroup(343,3)

Note that we don't need to compute $$p^3$$; we can also write this as:

SmallGroup(7^3,3)

As an extraspecial group
For any prime $$p$$, we can define this group using GAP's ExtraspecialGroup function as:

ExtraspecialGroup(p^3,'+')

For $$p \ne 2$$, it can also be constructed as:

ExtraspecialGroup(p^3,p)

where the argument $$p$$ indicates that it is the extraspecial group of exponent $$p$$. For instance, for $$p = 5$$:

ExtraspecialGroup(5^3,5)