Dicyclic group

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with metacyclic group

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with dihedral group

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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:

It is given by the presentation:

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

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:

The identity element. (1)
The unique central non-identity element, which is given by $a^n = x^2$ . (1)
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.
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.