Dicyclic group

Dicyclic group

In group theory, a dicyclic group (notation Dicn) is a member of a class of non-abelian groups 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:

1 \to C_{2n} \to \mbox{Dic}_n \to C_2 \to 1. \,

More generally, given any finite abelian group with an order-2 element, one can define a dicyclic group.

Contents

Definition

For each integer n > 1, the dicyclic group Dicn can be defined as the subgroup of the unit quaternions generated by

\begin{align} a & = e^{i\pi/n} = \cos\frac{\pi}{n} + i\sin\frac{\pi}{n} \\
                      x & = j
       \end{align}

More abstractly, one can define the dicyclic group Dicn as any group having the presentation

\mbox{Dic}_n = \langle a,x \mid a^{2n} = 1,\ x^2 = a^n,\ x^{-1}ax = a^{-1}\rangle.\,\!

Some things to note which follow from this definition:

  • x4 = 1
  • x2ak = ak+n = akx2
  • if j = ±1, then xjak = a-kxj.
  • akx−1 = aknanx−1 = aknx2x−1 = aknx.

Thus, every element of Dicn can be uniquely written as akxj, where 0 ≤ k < 2n and j = 0 or 1. The multiplication rules are given by

  • akam = ak + m
  • akamx = ak + mx
  • akxam = akmx
  • akxamx = akm + n

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.

Properties

For each n > 1, the dicyclic group Dicn is a non-abelian group of order 4n. ("Dic1" is C4, the cyclic group of order 4, which is abelian, and is not considered dicyclic.)

Let A = <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.

Binary dihedral group

Dicyclic-commutative-diagram.svg

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.

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, Dicn is not a semidirect product of A and <x>, since A ∩ <x> is not trivial.

The dicyclic group has a unique involution (i.e. an element of order 2), namely x2 = an. Note that this element lies in the center of Dicn. Indeed, the center consists solely of the identity element and x2. If we add the relation x2 = 1 to the presentation of Dicn one obtains a presentation of the dihedral group Dih2n, so the quotient group Dicn/<x2> is isomorphic to Dihn.

There is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation group described at quaternions and spatial rotations. Since the dicyclic group can be embedded inside the unit quaternions one can ask what the image of it is under this homomorphism. The answer is the just the dihedral symmetry group Dihn. For this reason the dicyclic group is also known as the binary dihedral group. Note that the dicyclic group does not contain any subgroup isomorphic to Dihn.

The analogous pre-image construction, using Pin+(2) instead of Pin(2), yields another dihedral group, Dih2n, rather than a dicyclic group.

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.

See also

References

  • Coxeter, H. S. M. (1974), "7.1 The Cyclic and Dicyclic groups", Regular Complex Polytopes, Cambridge University Press, pp. 74–75 .

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Pin group — In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2 to 1 to the orthogonal group, just as the spin group maps 2 to 1 to the special orthogonal group.In general the map from the… …   Wikipedia

  • Quaternion group — In group theory, the quaternion group is a non abelian group of order 8. It is often denoted by Q and written in multiplicative form, with the following 8 elements : Q = {1, −1, i , − i , j , − j , k , − k }Here 1 is the identity element, (−1)2 …   Wikipedia

  • Dihedral group — This snowflake has the dihedral symmetry of a regular hexagon. In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections.[1] Dihedr …   Wikipedia

  • List of group theory topics — Contents 1 Structures and operations 2 Basic properties of groups 2.1 Group homomorphisms 3 Basic types of groups …   Wikipedia

  • Binary icosahedral group — In mathematics, the binary icosahedral group is an extension of the icosahedral group I of order 60 by a cyclic group of order 2. It can be defined as the preimage of the icosahedral group under the 2:1 covering homomorphism:mathrm{Sp}(1) o… …   Wikipedia

  • Metacyclic group — In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group G for which there is a short exact sequence where H and K are cyclic. Equivalently, a metacyclic group is a group G having a cyclic… …   Wikipedia

  • Point groups in three dimensions — In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries… …   Wikipedia

  • List of abstract algebra topics — Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this …   Wikipedia

  • List of small groups — The following list in mathematics contains the finite groups of small order up to group isomorphism.The list can be used to determine which known group a given finite group G is isomorphic to: first determine the order of G , then look up the… …   Wikipedia

  • List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”