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In group theory, a dicyclic group is a member of a class of groups Dic_{n} (n > 1), a non-abelian group of order 4n, which is an extension of the cyclic group of order 2
by a cyclic group (of even order $2n$), giving the name di-cyclic:
## Definition

_{n} as any group having the presentation
## Properties

## Binary dihedral group

## Generalizations

## See also

- $1\; to\; C\_\{2n\}\; to\; mbox\{Dic\}\_n\; to\; C\_2\; to\; 1$

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

- $a\; =\; e^\{ipi/n\}\; =\; cosfrac\{pi\}\{n\}\; +\; isinfrac\{pi\}\{n\}$

- $x\; =\; j,$

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

Some things to note which follow from this definition:

- x
^{4}= 1 - x
^{2}a^{k}= a^{k+n}= a^{k}x^{2} - if j = ±1, then x
^{j}a^{k}= a^{-k}x^{j}. - a
^{k}x^{−1}= a^{k−n}a^{n}x^{−1}= a^{k−n}x^{2}x^{−1}= a^{k−n}x.

Thus, every element of Dic_{n} can be uniquely written as a^{k}x^{j}, where 0 ≤ k < 2n and j = 0 or 1. The multiplication rules are given by

- $a^k\; a^m\; =\; a^\{k+m\}$
- $a^k\; a^m\; x\; =\; a^\{k+m\}x$
- $a^k\; x\; a^m\; =\; a^\{k-m\}x$
- $a^k\; x\; a^m\; x\; =\; a^\{k-m+n\}$

It follows that Dic_{n} 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 Dic_{n} is a non-abelian group of order 4n. ("Dic_{1}" is $C\_4$, the cyclic group of order 4, which is abelian, and is not considered dicyclic.)

Let A = <a> be the subgroup of Dic_{n} generated by a. Then A is a cyclic group of order 2n, so [Dic_{n}:A] = 2. As a subgroup of index 2 it is automatically a normal subgroup. The quotient group Dic_{n}/A is a cyclic group of order 2.

Dic_{n} 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 $operatorname\{Pin\}\_-(2)$—and in this context is known as the binary dihedral group.

The connection with the binary cyclic group $C\_\{2n\}$, the cyclic group $C\_n$, and the dihedral group Dih_{n} 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 x^{2} = 1, instead of x^{2} = a^{n}; and this yields a different structure. In particular, Dic_{n} 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 x^{2} = a^{n}. Note that this element lies in the center of Dic_{n}. Indeed, the center consists solely of the identity element and x^{2}. If we add the relation x^{2} = 1 to the presentation of Dic_{n} one obtains a presentation of the dihedral group Dih_{2n}, so the quotient group Dic_{n}/<x^{2}> is isomorphic to Dih_{n}.

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 Dih_{n}. 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 Dih_{n}.

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, x^{2} = y, and for all a in A, x^{-1}ax = 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.

- binary polyhedral group
- binary cyclic group
- binary tetrahedral group
- binary octahedral group
- binary icosahedral group

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Last updated on Saturday May 31, 2008 at 02:17:53 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday May 31, 2008 at 02:17:53 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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