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).
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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-1akx = (x-1ax)k = a-k.
- x has order 4 in Dic(A)
- x2ak = ak+n = akx2
- if j = ±1, then xjak = a-kxj.
- akx-1 = ak-nanx-1 = ak-nx2x-1 = ak-nx.
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.
