Binary Dihedral Groups#
AUTHORS:
Travis Scrimshaw (2016-02): initial version
- class sage.groups.matrix_gps.binary_dihedral.BinaryDihedralGroup(n)#
Bases:
UniqueRepresentation
,FinitelyGeneratedMatrixGroup_gap
The binary dihedral group \(BD_n\) of order \(4n\).
Let \(n\) be a positive integer. The binary dihedral group \(BD_n\) is a finite group of order \(4n\), and can be considered as the matrix group generated by
\[\begin{split}g_1 = \begin{pmatrix} \zeta_{2n} & 0 \\ 0 & \zeta_{2n}^{-1} \end{pmatrix}, \qquad\qquad g_2 = \begin{pmatrix} 0 & \zeta_4 \\ \zeta_4 & 0 \end{pmatrix},\end{split}\]where \(\zeta_k = e^{2\pi i / k}\) is the primitive \(k\)-th root of unity. Furthermore, \(BD_n\) admits the following presentation (note that there is a typo in [Sun2010]):
\[BD_n = \langle x, y, z | x^2 = y^2 = z^n = x y z \rangle.\](The \(x\), \(y\) and \(z\) in this presentations correspond to the \(g_2\), \(g_2 g_1^{-1}\) and \(g_1\) in the matrix group avatar.)
REFERENCES:
- cardinality()#
Return the order of
self
, which is \(4n\).EXAMPLES:
sage: G = groups.matrix.BinaryDihedral(3) sage: G.order() 12
- order()#
Return the order of
self
, which is \(4n\).EXAMPLES:
sage: G = groups.matrix.BinaryDihedral(3) sage: G.order() 12