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((Group theory sidebar |Finite))
((Refimprove|date=February 2010))
In [[group theory]], a '''dicyclic group''' (notation '''Dic'''<sub>''n''</sub> or '''Q'''<sub>4''n''</sub>,<ref>((cite book|last1=Nicholson|first1=W. Keith|title=Introduction to Abstract Algebra|date=1999|publisher=John Wiley & Sons, Inc.|location=New York|isbn=0-471-33109-0|page=449|edition=2nd))</ref> ((angbr|''n'',2,2))<ref name=anglebracket> Coxeter&Moser: Generators and Relations for discrete groups: <l,m,n>: R<sup>l</sup> = S<sup>m</sup> = T<sup>n</sup> = RST</ref>) is a particular kind of [[non-abelian group]] of [[Order (group theory)|order]] 4''n'' (''n'' > 1).  It is an [[group extension|extension]] of the [[cyclic group]] of order 2 by a cyclic group of order 2''n'', giving the name ''di-cyclic''. In the notation of [[exact sequence]]s of groups, this extension can be expressed as:

:<math>1 \to C_{2n} \to \mbox{Dic}_n \to C_2 \to 1. \, </math>

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

==Definition==

For each [[integer]] ''n'' > 1, the dicyclic group Dic<sub>''n''</sub> can be defined as the [[subgroup]] of the unit [[quaternion]]s generated by

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

More abstractly, one can define the dicyclic group Dic<sub>''n''</sub> as the group with the following [[presentation of a group|presentation]]<ref name="Roman">((cite book |last1=Roman |first1=Steven |authorlink1=Steven Roman |title=Fundamentals of Group Theory: An Advanced Approach |year=2011 |publisher=Springer |isbn=9780817683016 |pages=347–348 ))</ref>

:<math>\operatorname{Dic}_n = \left\langle a, x \mid a^{2n} = 1,\ x^2 = a^n,\ x^{-1}ax = a^{-1}\right\rangle.\,\!</math>

Some things to note which follow from this definition:
*<math> x^4 = 1 </math>
*<math> x^2 a^m = a^{m+n} = a^m x^2 </math>
*if <math> l = \pm 1 </math>, then <math> x^l a^m = a^{-m} x^l </math>
*<math> a^m x^{-1}= a^{m-n} a^n x^{-1}= a^{m-n} x^2 x^{-1}= a^{m-n} x </math>

Thus, every element of Dic<sub>''n''</sub> can be uniquely written as ''a''<sup>''m''</sup>''x''<sup>''l''</sup>, where 0 ≤ ''m'' < 2''n'' and ''l'' = 0 or 1. The multiplication rules are given by
*<math>a^k a^m = a^{k+m}</math>
*<math>a^k a^m x = a^{k+m}x</math>
*<math>a^k x a^m = a^{k-m}x</math>
*<math>a^k x a^m x = a^{k-m+n}</math>
It follows that Dic<sub>''n''</sub> has [[order (group theory)|order]] 4''n''.<ref name="Roman"/>

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]].<ref name="Roman"/>

==Properties==

For each ''n'' > 1, the dicyclic group Dic<sub>''n''</sub> is a [[non-abelian group]] of order 4''n''.  (For the degenerate case ''n'' = 1, the group Dic<sub>1</sub> is the cyclic group ''C''<sub>4</sub>, which is not considered dicyclic.)

Let ''A'' = ((angbr|''a'')) be the subgroup of Dic<sub>''n''</sub> [[generating set of a group|generated]] by ''a''. Then ''A'' is a cyclic group of order 2''n'', so [Dic<sub>''n''</sub>:''A''] = 2. As a subgroup of [[Index of a subgroup|index]] 2 it is automatically a [[normal subgroup]]. The quotient group Dic<sub>''n''</sub>/''A'' is a cyclic group of order 2.

Dic<sub>''n''</sub> is [[solvable group|solvable]]; note that ''A'' is normal, and being abelian, is itself solvable.

==Binary dihedral group==
[[Image:Dicyclic-commutative-diagram.svg|right|400px]]

The dicyclic group is a [[binary polyhedral group]] — it is one of the classes of subgroups of the [[Pin group]] Pin<sub>−</sub>(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]] ''C''<sub>2''n''</sub>, the cyclic group ''C''<sub>''n''</sub>, and the [[dihedral group]] Dih<sub>''n''</sub> of order 2''n'' 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''⟩.

There is a superficial resemblance between the dicyclic groups and [[dihedral group]]s; both are a sort of "mirroring" of an underlying cyclic group. But the presentation of a dihedral group would have ''x''<sup>2</sup> = 1, instead of ''x''<sup>2</sup> = ''a''<sup>''n''</sup>; and this yields a different structure. In particular, Dic<sub>''n''</sub> is not a [[semidirect product]] of ''A'' and ((angbr|''x'')), since ''A''&nbsp;∩&nbsp;((angbr|''x'')) is not trivial.

The dicyclic group has a unique [[involution (mathematics)#Group theory|involution]] (i.e. an element of order 2), namely ''x''<sup>2</sup> = ''a''<sup>''n''</sup>. Note that this element lies in the [[center of a group|center]] of Dic<sub>''n''</sub>. Indeed, the center consists solely of the identity element and ''x''<sup>2</sup>. If we add the relation ''x''<sup>2</sup> = 1 to the presentation of Dic<sub>''n''</sub> one obtains a presentation of the [[dihedral group]] Dih<sub>''n''</sub>, so the quotient group Dic<sub>''n''</sub>/<''x''<sup>2</sup>> is isomorphic to Dih<sub>''n''</sub>.

There is a natural 2-to-1 [[homomorphism]] from the group of unit quaternions to the 3-dimensional [[rotation group SO(3)|rotation group]] described at [[quaternions and spatial rotation]]s. 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 just the dihedral symmetry group Dih<sub>''n''</sub>. 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<sub>''n''</sub>.

The analogous pre-image construction, using Pin<sub>+</sub>(2) instead of Pin<sub>−</sub>(2), yields another dihedral group, Dih<sub>2''n''</sub>, 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, ''x''<sup>2</sup> = ''y'', and for all ''a'' in ''A'', ''x''<sup>−1</sup>''ax'' = ''a''<sup>−1</sup>.

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.

The dicyclic group is the case <math>(p,q,r)=(2,2,n) </math> of the family of binary triangle groups <math>\Gamma(p,q,r)</math> defined by the presentation:[https://groupprops.subwiki.org/wiki/Binary_von_Dyck_group]<blockquote><math>\langle a,b,c \mid a^p = b^q = c^r = abc \rangle.</math></blockquote>Taking the quotient by the additional relation <math>abc = e</math> produces an ordinary [[triangle group]], which in this case is the dihedral quotient <math>\mathrm{Dic}_n\rightarrow  \mathrm{Dih}_n</math>.

==See also==
*[[binary polyhedral group]]
*[[binary cyclic group]], ⟨''n''⟩, order 2''n''
*[[binary tetrahedral group]], 2T = ⟨2,3,3⟩,<ref name=anglebracket/> order 24
*[[binary octahedral group]], 2O = ⟨2,3,4⟩,<ref name=anglebracket/> order 48
*[[binary icosahedral group]], 2I = ⟨2,3,5⟩,<ref name=anglebracket/> order 120

==References==
((Reflist))
((refbegin))
* ((Citation | authorlink = Harold Scott MacDonald Coxeter | last = Coxeter | first = H. S. M. | title = Regular Complex Polytopes | publisher = Cambridge University Press | year = 1974 | chapter = 7.1 The Cyclic and Dicyclic groups | pages = [https://books.google.com/books?id=9BY9AAAAIAAJ&pg=PA74 74–75] )).
*((cite book |author1=Coxeter, H. S. M.  |author2=Moser, W. O. J.  | title=Generators and Relations for Discrete Groups | location=New York | publisher=Springer-Verlag | year=1980 | isbn=0-387-09212-9))
((refend))

==External links==
* [http://groupnames.org/#?dicyclic Dicyclic groups on GroupNames]

((DEFAULTSORT:Dicyclic Group))
[[Category:Finite groups]]
[[Category:Quaternions]]

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