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Unitriangular matrix group:UT(3,p)

524 bytes removed, 16:02, 28 March 2012
Definition
===As a group of matrices===
Given a prime <math>p</math>, the group <math>UUT(3,p)</math> is defined as follows: it is the [[unitriangular matrix group ]] of upper triangular matrices with 1s on the diagonal, and entries [[unitriangular matrix group of degree three|degree three]] over <math>F_p</math> (with the group operation being matrix multiplication). Each such matrix <math>(a_{ij})</math> can be described by the three entries [[prime field]] <math>a_\mathbb{12}, a_{13}, a_{23F}_p</math>. The matrix looks like: <math>\begin{pmatrix}1 & a_{12} & a_{13} \\0 & 1 & a_{23}\\0 & 0 & 1\end{pmatrix}</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 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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