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 BDn of order 4n.

Let n be a positive integer. The binary dihedral group BDn is a finite group of order 4n, and can be considered as the matrix group generated by

g1=(ζ2n00ζ12n),g2=(0ζ4ζ40),

where ζk=e2πi/k is the primitive k-th root of unity. Furthermore, BDn admits the following presentation (note that there is a typo in [Sun2010]):

BDn=x,y,z|x2=y2=zn=xyz.

(The x, y and z in this presentations correspond to the g2, g2g11 and g1 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