Binary Dihedral Groups

AUTHORS:

  • Travis Scrimshaw (2016-02): initial version
class sage.groups.matrix_gps.binary_dihedral.BinaryDihedralGroup(n)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.groups.matrix_gps.finitely_generated.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 [Sun]):

\[BD_n = \langle x, y, z | x^2 = y^2 = z^n = x y z \rangle.\]

(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:

[Dolgachev09]Igor Dolgachev. McKay Correspondence. (2009). http://www.math.lsa.umich.edu/~idolga/McKaybook.pdf
[Sun]Yi Sun. The McKay correspondence. http://www.math.miami.edu/~armstrong/686sp13/McKay_Yi_Sun.pdf
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