Dicyclic group

In group theory, a dicyclic group (notation Dicn or Q4n, ) is a particular kind of non-abelian group of order 4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di-cyclic. In the notation of exact sequences of groups, this extension can be expressed as: More generally, given any finite abelian group with an order-2 element, one can define a dicyclic group. For each integer n > 1, the dicyclic group Dicn can be defined as the subgroup of the unit quaternions generated by More abstractly, one can define the dicyclic group Dicn as the group with the following presentation Some things to note which follow from this definition: if , then Thus, every element of Dicn can be uniquely written as amxl, where 0 ≤ m < 2n and l = 0 or 1. The multiplication rules are given by It follows that Dicn has order 4n. When n = 2, the dicyclic group is isomorphic to the quaternion group Q. More generally, when n is a power of 2, the dicyclic group is isomorphic to the generalized quaternion group. For each n > 1, the dicyclic group Dicn is a non-abelian group of order 4n. (For the degenerate case n = 1, the group Dic1 is the cyclic group C4, which is not considered dicyclic.) Let A = be the subgroup of Dicn generated by a. Then A is a cyclic group of order 2n, so [Dicn:A] = 2. As a subgroup of index 2 it is automatically a normal subgroup. The quotient group Dicn/A is a cyclic group of order 2. Dicn is solvable; note that A is normal, and being abelian, is itself solvable. The dicyclic group is a binary polyhedral group — it is one of the classes of subgroups of the Pin group Pin−(2), which is a subgroup of the Spin group Spin(3) — and in this context is known as the binary dihedral group. The connection with the binary cyclic group C2n, the cyclic group Cn, and the dihedral group Dihn of order 2n is illustrated in the diagram at right, and parallels the corresponding diagram for the Pin group. Coxeter writes the binary dihedral group as ⟨2,2,n⟩ and binary cyclic group with angle-brackets, ⟨n⟩.

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