DihedralGroup - Maple Help

GroupTheory

 DihedralGroup
 construct a dihedral group of a given degree

 Calling Sequence DihedralGroup( n ) DihedralGroup( n, s )

Parameters

 n - : algebraic : an expression understood to be a positive integer or $\mathrm{\infty }$ s - : equation : (optional) equation of the form form = "fpgroup" or form = "permgroup" (default)

Description

 • The dihedral group of degree $n$ is the symmetry group of an $n$-sided regular polygon for $n>2$. It is generated by a reflection (of order $2$), and a rotation (of order $n$). It acts as a permutation group on the vertices of the regular $n$-sided polygon.
 • For $n=1$, the dihedral group is a cyclic group of order $2$.  For $n=2$, the dihedral group is the non-cyclic group of order $4$, also known as the Klein $4$-group.
 • If $n=\mathrm{\infty }$, then an infinite dihedral group (a free product of two groups of order two, or the holomorph of an infinite cyclic group) is returned as a finitely presented group.
 • The DihedralGroup( n ) command returns a dihedral group, either as a permutation group or a group defined by generators and defining relations. By default, if n is a positive integer, then a permutation group is returned, but a finitely presented group can be requested by passing the option 'form' = "fpgroup". If $n=\mathrm{\infty }$ then a finitely presented group is returned, regardless of any form option passed.
 • If the value of the parameter n is not numeric, then a symbolic group representing the dihedral group of the indicated degree is returned.
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{DihedralGroup}\left(13\right)$
 ${G}{≔}{{\mathbf{D}}}_{{13}}$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${26}$ (2)
 > $G≔\mathrm{DihedralGroup}\left(13,\mathrm{form}="fpgroup"\right)$
 ${G}{≔}{{\mathbf{D}}}_{{13}}$ (3)
 > $G≔\mathrm{DihedralGroup}\left(17,\mathrm{form}="permgroup"\right)$
 ${G}{≔}{{\mathbf{D}}}_{{17}}$ (4)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${34}$ (5)
 > $\mathrm{AreIsomorphic}\left(\mathrm{DihedralGroup}\left(3\right),\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{GroupOrder}\left(\mathrm{DihedralGroup}\left(3k\right)\right)$
 ${6}{}{k}$ (7)
 > $\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(6k\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}k::\mathrm{posint}$
 ${\mathrm{false}}$ (8)
 > $\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left({2}^{a}{4}^{b}\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{posint}$
 ${\mathrm{true}}$ (9)
 > $\mathrm{IsFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(7\right)\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{IsFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)$
 ${\mathrm{false}}$ (11)
 > $\mathrm{DrawCayleyTable}\left(\mathrm{DihedralGroup}\left(5\right),'\mathrm{conjugacy}'=\mathrm{true}\right)$
 > $\mathrm{ClassNumber}\left(\mathrm{DihedralGroup}\left(6n\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}n::\mathrm{posint}$
 ${3}{}{n}{+}{3}$ (12)
 > $\mathrm{Exponent}\left(\mathrm{DihedralGroup}\left(2n+1\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}n::\mathrm{posint}$
 ${4}{}{n}{+}{2}$ (13)
 > $\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{DihedralGroup}\left(9\right)\right)$
 ${\mathrm{true}}$ (14)
 > $\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{DihedralGroup}\left(10\right)\right)$
 ${\mathrm{false}}$ (15)
 > $G≔\mathrm{DihedralGroup}\left(\mathrm{∞}\right)$
 ${G}{≔}{{\mathbf{D}}}_{{\infty }}$ (16)
 > $\mathrm{IsNilpotent}\left(G\right)$
 ${\mathrm{false}}$ (17)
 > $\mathrm{IsSupersoluble}\left(G\right)$
 ${\mathrm{true}}$ (18)
 > $\mathrm{IdentifyFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(11\right)\right)$
 ${22}{,}{1}$ (19)
 > $\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{DihedralGroup}\left(5\right)\right)\right)$
 insertdirect, content = "C1a2a5a5b|C|1522LUkjbWlHNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EnY2hpX18xNiI=LUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EiMTYiLUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EiMTYiLUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N\     5c2xpYkc2I1EiMTYiLUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EiMTYiLUkjbWlHNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EnY2hpX18yNiI=LUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EiMTYiLUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1ErJnVtaW51czA7MTYiLUkjbW5HNiQlKnByb3RlY3RlZEcvJSt\     tb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EiMTYiLUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EiMTYiLUkjbWlHNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EnY2hpX18zNiI=LUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EiMjYiLUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EiMDYiLUklbXJvd0c2JCUqcHJvdGVjdGVkRy8lK21vZHVsZW5hbWVHSSxUeXBlc2V0dGluZ0c2JEYlJShfc3lzbGliRzYlLUklbXN1cEc2JEYlL0YnRig2JC1JKG1mZW5jZWRHNiRGJS9GJ0YoNiMtSSNtbkc2JEYlL0YnRig2I1ErJnVtaW51czA7MTYiLUkmbWZyYWNHNiRGJS9GJ0YoNiUtRjc2I1EiMkY8LUY3NiNRIjVGPC8lKWJldmVsbGVkR1EldHJ1ZUY8LUkjbW9HNiRGJS9GJ0YoNiNRKCZtaW51cztGPC1GLTYkRjEtRj42JS1GNzYjUSIzRjxGRUZILUklbXJvd0c2JCUqcHJvdGVjdGVkRy8lK21vZHVsZW5hbWVHSSxUeXBlc2V0dGluZ0c2JEYlJShfc3lzbGliRzYnLUklbXN1cEc2JEYlL0YnRig2JC1JKG1mZW5jZWRHNiRGJS9GJ0YoNiMtSSNtbkc2JEYlL0YnRig2I1ErJnVtaW51czA7MTYiLUkmbWZyYWNHNiRGJS9GJ0YoNiUtRjc2I1EiM0Y8LUY3NiNRIjVGPC8lKWJldmVsbGVkR1EldHJ1ZUY8LUkjbW9HNiRGJS9GJ0YoNiNRKCZtaW51cztGPC1GLTYkRjEtRj42JS1GNzYjUSIyRjxGRUZIRkstRjc2I1EiMUY8LUkjbWlHNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EnY2hpX180NiI=LUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EiMjYiLUkjbW5HNiQlKnByb3RlY3RlZEcvJSttb2R1bGVuYW1lR0ksVHlwZXNldHRpbmdHNiRGJSUoX3N5c2xpYkc2I1EiMDYiLUklbXJvd0c2JCUqcHJvdGVjdGVkRy8lK21vZHVsZW5hbWVHSSxUeXBlc2V0dGluZ0c2JEYlJShfc3lzbGliRzYnLUklbXN1cEc2JEYlL0YnRig2JC1JKG1mZW5jZWRHNiRGJS9GJ0YoNiMtSSNtbkc2JEYlL0YnRig2I1ErJnVtaW51czA7MTYiLUkmbWZyYWNHNiRGJS9GJ0YoNiUtRjc2I1EiM0Y8LUY3NiNRIjVGPC8lKWJldmVsbGVkR1EldHJ1ZUY8LUkjbW9HNiRGJS9GJ0YoNiNRKCZtaW51cztGPC1GLTYkRjEtRj42JS1GNzYjUSIyRjxGRUZIRkstRjc2I1EiMUY8LUklbXJvd0c2JCUqcHJvdGVjdGVkRy8lK21vZHVsZW5hbWVHSSxUeXBlc2V0dGluZ0c2JEYlJShfc3lzbGliRzYlLUklbXN1cEc2JEYlL0YnRig2JC1JKG1mZW5jZWRHNiRGJS9GJ0YoNiMtSSNtbkc2JEYlL0YnRig2I1ErJnVtaW51czA7MTYiLUkmbWZyYWNHNiRGJS9GJ0YoNiUtRjc2I1EiMkY8LUY3NiNRIjVGPC8lKWJldmVsbGVkR1EldHJ1ZUY8LUkjbW9HNiRGJS9GJ0YoNiNRKCZtaW51cztGPC1GLTYkRjEtRj42JS1GNzYjUSIzRjxGRUZI", state = "", minimal = true
 > $\mathrm{caygr}≔\mathrm{CayleyGraph}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{caygr}}{≔}{\mathrm{Graph 1: a directed graph with 8 vertices and 16 arc\left(s\right)}}$ (20)
 > $\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{caygr}\right)$

Compatibility

 • The GroupTheory[DihedralGroup] command was introduced in Maple 17.