Coverage for local/lib/python2.7/site-packages/sage/groups/matrix_gps/binary_dihedral.py : 94%

""" 

Binary Dihedral Groups 

AUTHORS: 

- Travis Scrimshaw (2016-02): initial version 

""" 

#***************************************************************************** 

# Copyright (C) 2016 Travis Scrimshaw <tscrimsh at umn.edu> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.groups.matrix_gps.finitely_generated import FinitelyGeneratedMatrixGroup_gap 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.misc.latex import latex 

from sage.rings.number_field.number_field import CyclotomicField 

from sage.matrix.matrix_space import MatrixSpace 

from sage.categories.groups import Groups 

from sage.rings.all import ZZ 

class BinaryDihedralGroup(UniqueRepresentation, FinitelyGeneratedMatrixGroup_gap): 

r""" 

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 

.. MATH:: 

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}, 

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]_): 

.. MATH:: 

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 

- :wikipedia:`Dicyclic_group#Binary_dihedral_group` 

""" 

def __init__(self, n): 

""" 

Initialize ``self``. 

EXAMPLES:: 

sage: G = groups.matrix.BinaryDihedral(4) 

sage: TestSuite(G).run() 

""" 

self._n = n 

if n % 2 == 0: 

R = CyclotomicField(2*n) 

zeta = R.gen() 

i = R.gen()**(n//2) 

else: 

R = CyclotomicField(4*n) 

zeta = R.gen()**2 

i = R.gen()**n 

MS = MatrixSpace(R, 2) 

zero = R.zero() 

gens = [ MS([zeta, zero, zero, ~zeta]), MS([zero, i, i, zero]) ] 

from sage.libs.gap.libgap import libgap 

gap_gens = [libgap(matrix_gen) for matrix_gen in gens] 

gap_group = libgap.Group(gap_gens) 

FinitelyGeneratedMatrixGroup_gap.__init__(self, ZZ(2), R, gap_group, category=Groups().Finite()) 

def _repr_(self): 

""" 

Return a string representation of ``self``. 

EXAMPLES:: 

sage: groups.matrix.BinaryDihedral(3) 

Binary dihedral group of order 12 

""" 

return "Binary dihedral group of order {}".format(4 * self._n) 

def _latex_(self): 

r""" 

Return a latex representation of ``self``. 

EXAMPLES:: 

sage: G = groups.matrix.BinaryDihedral(3) 

sage: latex(G) 

BD_{3} 

""" 

return "BD_{{{}}}".format(self._n) 

def order(self): 

""" 

Return the order of ``self``, which is `4n`. 

EXAMPLES:: 

sage: G = groups.matrix.BinaryDihedral(3) 

sage: G.order() 

12 

TESTS:: 

sage: for i in range(1, 10): 

....: G = groups.matrix.BinaryDihedral(5) 

....: assert len(list(G)) == G.order() 

""" 

return ZZ(4 * self._n) 

cardinality = order