Dicyclic group

From Groupprops
Jump to: navigation, search
WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with metacyclic group
WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with dihedral group
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a family of groups parametrized by the natural numbers, viz, for each natural number, there is a unique group (upto isomorphism) in the family corresponding to the natural number. The natural number is termed the parameter for the group family

Definition

The dicyclic group, also called the binary dihedral group with parameter n is defined in the following equivalent ways:

\langle a,x \mid a^{2n}=e, x^2 = a^n, xax^{-1} = a^{-1} \rangle

Here, e is the identity element.

  • It has the following faithful representation as a subgroup of the quaternions: a = e^{i\pi/n}, x = j.
  • It is the binary von Dyck group with parameters (n,2,2), i.e., it has the presentation:

\langle a,b,c \mid a^n = b^2 = c^2 = abc \rangle.

The dicyclic group with parameter n has order 4n, and it is an extension of a cyclic group of order 2n by a cyclic group of order 2.

Equivalence of definitions

Further information: equivalence of presentations of dicyclic group

Cases

The dicyclic group has some alternate descriptions in specific cases.

Case on n Examples Description of dicyclic group
Odd number n = 3, so dicyclic group:Dic12 Semidirect product of cyclic normal subgroup of order n (generated by a^2) and group of order 4 generated by x (in the first presentation) or b (in the second). The latter element conjugates a to its inverse.

Arithmetic functions

Here, the n is as in the parametrization. The order of the group is 4n.

Function Value Explanation
order 4n
exponent least common multiple of 4 and 2n
nilpotency class k + 1 if n = 2^k, undefined otherwise.
derived length 2 for n \ge 2
number of conjugacy classes n + 3
number of subgroups \sigma(n) + d(2n) where \sigma is the divisor sum function and d is the divisor count function

Group properties

Property Satisfied Explanation
Abelian group No for n \ge 2.
Nilpotent group Yes only for n a power of two.
Solvable group Yes
Supersolvable group Yes
Metacyclic group Yes
Ambivalent group Yes for n even, no for n odd
Rational group Yes only for n =2, i.e., the quaternion group

Particular cases

For small values

Note that all dicyclic groups are metacyclic and hence supersolvable. A dicyclic group is nilpotent if and only if it is of order 2^k for some k. It is abelian only if it has order 4.

Order of group Degree Common name for the group Comment
4 1 Cyclic group:Z4 Not typically considered a dicyclic group
8 2 Quaternion group
12 3 Dicyclic group:Dic12
16 4 Generalized quaternion group:Q16

Elements

Further information: Element structure of dicyclic groups

The dicyclic group of order 4n has n+3 conjugacy classes. In the discussion below, we use the presentation:

\langle a,x \mid a^{2n}=e, x^2 = a^n, x^{-1}ax = a^{-1} \rangle

The elements are:

  1. The identity element. (1)
  2. The unique central non-identity element, which is given by a^n = x^2. (1)
  3. The remaining elements in \langle a \rangle. There are 2n - 2 of these elements, and they occur in conjugacy classes of size two: each element is conjugate to its inverse. There are thus n - 1 conjugacy classes of size 2 each.
  4. The elements outside \langle a \rangle come in two conjugacy classes: the conjugacy class of x, which contains all elements of the form a^{2k}x, and the conjugacy class of ax. These two conjugacy classes are related by an outer automorphism and each has n elements.

Internal links