Statements (49)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:group_of_people
modular group discrete group |
| gptkbp:acts_by |
fractional linear transformations
|
| gptkbp:actsOn |
upper half-plane
|
| gptkbp:centralTo |
trivial
|
| gptkbp:definedIn |
PSL(2, Z) = SL(2, Z) / {±I}
|
| gptkbp:generation |
S
T |
| gptkbp:has_fundamental_domain |
{z in H : |z| ≥ 1, -1/2 ≤ Re(z) ≤ 1/2}
|
| gptkbp:hasProperty |
non-abelian
finitely presented free product of cyclic groups of order 2 and 3 |
| gptkbp:hasSubgroup |
gptkb:PSL(2,_R)
|
| https://www.w3.org/2000/01/rdf-schema#label |
modular group PSL(2, Z)
|
| gptkbp:importantFor |
complex analysis
modular forms number theory elliptic curves |
| gptkbp:loveInterest |
(ST)^3 = 1
S^2 = 1 |
| gptkbp:notable_quotient |
gptkb:PSL(2,_Z/NZ)
|
| gptkbp:notablePlace |
S
T |
| gptkbp:notation |
gptkb:PSL(2,_Z)
modular group Γ |
| gptkbp:order |
infinite
|
| gptkbp:presentedBy |
⟨S, T | S^2 = 1, (ST)^3 = 1⟩
|
| gptkbp:quotient_space |
modular group
|
| gptkbp:relatedTo |
modular arithmetic
modular functions modular group action modular group congruence subgroups modular symbols |
| gptkbp:S_matrix |
[[0, -1], [1, 0]]
|
| gptkbp:studiedBy |
gptkb:Felix_Klein
gptkb:Henri_Poincaré gptkb:Richard_Dedekind |
| gptkbp:T_matrix |
[[1, 1], [0, 1]]
|
| gptkbp:used_in |
gptkb:modular_curves
gptkb:string_theory gptkb:monstrous_moonshine automorphic forms modular functions modular forms theory modular group tessellation |
| gptkbp:bfsParent |
gptkb:modular_group
|
| gptkbp:bfsLayer |
5
|