Dicyclic group

In group theory, a dicyclic group is a member of a class of groupss which are formed by an extension of a group (generally a cyclic group) by a cyclic group of order 2 (the latter giving the name di-cyclic).

Table of contents

1 Definition
2 Properties
3 Generalizations

Definition

Let A = <a> be a cyclic group of even order 2n for n>1, generated by a. We define the dicyclic group Dic(A) as a group having a presentation with generators {a, x} and relations a²ⁿ =1, x² = aⁿ, and x^-1ax = a^-1.

Some things to note which follow from this definition:

A is a normal subgroup of Dic(A); since x^-1a^kx = (x^-1ax)^k = a^-k.
x has order 4 in Dic(A)
x²a^k = a^k+n = a^kx²
if j = ±1, then x^ja^k = a^-kx^j.
a^kx^-1 = a^k-naⁿx^-1 = a^k-nx²x^-1 = a^k-nx.

Thus, every element of Dic(A) can be uniquely written as a^kx^j, where j = 0 or 1; so [Dic(A):A] = 2, and |Dic(A)| = 2|A|.

If A has order which is a power of 2, then Dic(A) is called a generalized quaternion group; if A = C₄, then we get the quaternion group.

Properties

By its definition, a dicyclic group is always non-abelian (one doesn't consider "Dic(C₂)" as dicyclic).

There is a superficial resemblance between the dicyclic groups and dihedral groups; both are a sort of "mirroring" of an underlying cyclic group. But the presentation of a dihedral group would have x² = 1, instead of x² = aⁿ; and this yields a different structure. In particular, Dic(A) is not a semidirect product of A and <x>, since A ∩ <x> is not trivial. Instead, Dic(A) is a cyclic extension of A.

Dic(A) is solvable; note that A is normal, and being abelian, is itself solvable.

Generalizations

Let A be an abelian group, having a specific element y in A with order 2. A group G is called a generalized dicyclic group, written as Dic(A, y), if it is generated by A and an additional element x, and in addition we have that [G:A] = 2, x² = y, and for all a in A, x^-1ax = a^-1.

Since for a cyclic group of even order, there is always a unique element of order 2, we can see that dicyclic groups are just a specific type of generalized dicyclic group.

Generalized dicyclic groups, in turn, are examples of cyclic extensions.