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=(ζ2n00ζ−12n),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, g2g−11 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
-