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Modular group
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==Hecke groups== The modular group can be generalized to the '''Hecke groups''', named for [[Erich Hecke]], and defined as follows.<ref>{{cite book|title=Combinatorial Group Theory, Discrete Groups, and Number Theory|first1=Gerhard|last1=Rosenberger|first2=Benjamin|last2=Fine|first3=Anthony M.|last3=Gaglione|first4=Dennis|last4=Spellman|year=2006 |url=https://books.google.com/books?id=5Unmxs7yeHwC&pg=PA65|page=65|publisher=American Mathematical Society |isbn=9780821839850 }}</ref> The Hecke group {{math|''H''<sub>''q''</sub>}} with {{math|''q'' β₯ 3}}, is the discrete group generated by :<math>\begin{align} z &\mapsto -\frac1z \\ z &\mapsto z + \lambda_q, \end{align}</math> where {{math|''Ξ»<sub>q</sub>'' {{=}} 2 cos {{sfrac|Ο|''q''}}}}. For small values of {{math|''q'' β₯ 3}}, one has: :<math>\begin{align} \lambda_3 &= 1, \\ \lambda_4 &= \sqrt{2}, \\ \lambda_5 &= \frac{1+\sqrt{5}}{2}, \\ \lambda_6 &= \sqrt{3}, \\ \lambda_8 &= \sqrt{2+\sqrt{2}}. \end{align}</math> The modular group {{math|Ξ}} is isomorphic to {{math|''H''<sub>3</sub>}} and they share properties and applications β for example, just as one has the [[free product]] of [[cyclic group]]s :<math>\Gamma \cong C_2 * C_3,</math> more generally one has :<math>H_q \cong C_2 * C_q,</math> which corresponds to the [[triangle group]] {{math|(2, ''q'', β)}}. There is similarly a notion of principal congruence subgroups associated to principal ideals in {{math|'''Z'''[''Ξ»'']}}.
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