Editing Dicyclic group (section)

==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>.