# Binary Dihedral Groups¶

AUTHORS:

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

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