Unitriangular matrix group:UT(4,p)

Definition
The cases $$p = 2$$ (see unitriangular matrix group:UT(4,2)) and $$p = 3$$ (see unitriangular matrix group:UT(4,3)) differ somewhat from the cases of other primes. This is noted at all places in the page where it becomes significant.

As a group of matrices
Given a prime $$p$$, the group $$UT(4,p)$$ is defined as the unitriangular matrix group of degree four over the prime field $$\mathbb{F}_p$$. Explicitly, this is described as the following group under matrix multiplication:

$$\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} & a_{14}\\ 0 & 1 & a_{23} & a_{24} \\ 0 & 0 & 1 & a_{34}\\ 0 & 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{14},a_{23},a_{24},a_{34} \in \mathbb{F}_p \right \}$$

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


 * $$c_{12} = a_{12} + b_{12}$$
 * $$c_{13} = a_{13} + b_{13} + a_{12}b_{23}$$
 * $$c_{14} = a_{14} + b_{14} + a_{12}b_{24} + a_{13}b_{34}$$
 * $$c_{23} = a_{23} + b_{23}$$
 * $$c_{24} = a_{24} + b_{24} + a_{23}b_{34}$$
 * $$c_{34} = a_{34} + b_{34}$$

The identity element is the identity matrix, i.e., the matrix where all off-diagonal entries are zero and all diagonal entries are 1.

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


 * $$m_{12} = -a_{12}$$
 * $$m_{13} = -a_{13} + a_{12}a_{23}$$
 * $$m_{14} = -a_{14} + a_{12}a_{24} + a_{13}a_{34} - a_{12}a_{23}a_{34}$$
 * $$m_{23} = -a_{23}$$
 * $$m_{24} = -a_{24} + a_{23}a_{34}$$
 * $$m_{34} = -a_{34}$$

In coordinate form
We may define the group as the set of ordered 6-tuples $$(a_{12},a_{13},a_{14},a_{23},a_{24}, a_{34})$$ over $$\mathbb{F}_p$$ (the coordinates are allowed to repeat), with the multiplication law, identity element, and inverse operation given by:

$$ (a_{12},a_{13},a_{14},a_{23},a_{24},a_{34}) (b_{12},b_{13},b_{14},b_{23},b_{24},b_{34}) = $$

$$(a_{12} + b_{12},a_{13} + b_{13} + a_{12}b_{23},a_{14} + b_{14} + a_{12}b_{24} + a_{13}b_{34}, a_{23} + b_{23}, a_{24} + b_{24} + a_{23}b_{34},a_{34} + b_{34})$$

$$\mbox{Identity element} = (0,0,0,0,0,0)$$

$$(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})^{-1} = (-a_{12},-a_{13} + a_{12}a_{23}, -a_{14} + a_{12}a_{24} + a_{13}a_{34} - a_{12}a_{23}a_{34}, -a_{23}, -a_{34}, -a_{24} + a_{23}a_{34})$$

The matrix corresponding to the 6-tuple $$(a_{12},a_{13},a_{14},a_{23},a_{24}, a_{34})$$ is:

$$\begin{pmatrix} 1 & a_{12} & a_{13} & a_{14}\\ 0 & 1 & a_{23} & a_{24} \\ 0 & 0 & 1 & a_{34}\\ 0 & 0 & 0 & 1 \\\end{pmatrix}$$

This definition clearly matches the earlier definition, based on the rules of matrix multiplication.

Families
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 $$GL(n,q)$$.

GAP implementation
We assume $$p$$ is assigned a prime number value beforehand.