ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 27 Jan 2022 23:55:08 GMT2022-01-27T23:55:08Z50211- Characterisations of Ap+2 and L2 (2q)https://scholarbank.nus.edu.sg/handle/10635/102975Title: Characterisations of Ap+2 and L2 (2q)
Authors: Lang, M.-L.
Abstract: Let G be a transitive group of degree p + 2 with p ∥ G | where p ≧ 5 is a prime number, then (i) G is isomorphic to Sp+2 or Ap+2, if G has an element of order 4, (ii) G is isomorphic to L2(2q) or P Γ L2(2q), if 2q - 1 = p is a Mersenne prime and G has no element of order 4.
Fri, 02 May 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1029751997-05-02T00:00:00Z
- Congruence subgroups associated to the monsterhttps://scholarbank.nus.edu.sg/handle/10635/103041Title: Congruence subgroups associated to the monster
Authors: Chua, K.S.; Lang, M.L.
Abstract: Let Δ = {G : g(G) = 0,Γ0(m) ≤ G ≤ N(Γ0(m)) for some m}, where N(Γ0(m)) is the normaliser of Γ0(m) in PSL2(ℝ) and g(G) is the genus of ℍ*/G. In this article, we determine all the m. Further, for each m, we list all the intermediate groups G of Γ0(m) ≤ N(Γ0(m)) such that g(G) - 0. All the intermediate groups of width 1 at ∞ are also listed in a separate table (see www.math.nus.edu.sg/ ~matlml/).
Thu, 01 Jan 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030412004-01-01T00:00:00Z
- Transitive groups of degree 2p + khttps://scholarbank.nus.edu.sg/handle/10635/104395Title: Transitive groups of degree 2p + k
Authors: Lang, M.-L.
Abstract: Let p be an odd prime. We study transitive groups of degree n = 2p + k where k is 1, 2 or p. A transitive group G (|G| even) of degree 2p + 1 is doubly transitive of degree 2p + 1 if and only if G admits an element of order p degree 2p. A primtive group G (|Ga| even) of degree 2p + 2 is triply transitive of degree 2p + 2 if and only G admits an element of order p degree 2p. A primitive group G (|G| even) of degree 3p with rank less than 4 is doubly transitive if p is not of the form 3a2 + 3a + 1 or (3b2 + 3b + 2)/4.
Mon, 04 May 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043951998-05-04T00:00:00Z
- The signatures of the congruence subgroups G0(τ) of the hecke groups G4 and G6https://scholarbank.nus.edu.sg/handle/10635/104353Title: The signatures of the congruence subgroups G0(τ) of the hecke groups G4 and G6
Authors: Lang, M.-L.
Abstract: We determine the signatures of the congruence subgroups of the Hecke groups G4 and G6.
Sat, 01 Jan 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043532000-01-01T00:00:00Z
- Principal Congruence Subgroups of the Hecke Groupshttps://scholarbank.nus.edu.sg/handle/10635/103972Title: Principal Congruence Subgroups of the Hecke Groups
Authors: Lang, M.-L.; Lim, C.-H.; Tan, S.-P.
Abstract: Let q be an odd integer >3 and let Gq be the Hecke group associated to q. Let (τ) be a prime ideal of Z[λq] and G(q, τ) the principal congruence subgroup of Gq associated to τ. We give a formula for [Gq:G(q, τ)], the index of the principal congruence subgroup G(q, τ) in Gq. We also give formulae for the indices [G1(q, τ), G(q, τ)] and [G0(q, τ), G1(q, τ)]. Finally, we give a formula for the geometric invariants of G(q, τ) when q is a rational prime. © 2000 Academic Press.
Fri, 01 Dec 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039722000-12-01T00:00:00Z
- Normalizers of the congruence subgroups of the Hecke group G5https://scholarbank.nus.edu.sg/handle/10635/103642Title: Normalizers of the congruence subgroups of the Hecke group G5
Authors: Lang, M.-L.; Tan, S.-P.
Abstract: Let A = 2cos(π/5) and let G be the Hecke, group associated to A.In this article, we show that for τ a prime ideal in ℤ [λ], the congruence subgroups G0(τ) of G are self-normalized in P5L2(ℝ).© 1999 American Mathematical Society.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036421999-01-01T00:00:00Z
- Normalizers of the congruence subgroups of the hecke group G5 IIhttps://scholarbank.nus.edu.sg/handle/10635/103643Title: Normalizers of the congruence subgroups of the hecke group G5 II
Authors: Lang, M.-L.; Tan, S.-P.
Abstract: Let λ = 2 cos(π/5). Let (τ) be an ideal of ℤ[λ] and let (τ0) be the maximal ideal of ℤ[λ] such that (τ0 2) ⊆ (τ). Then N(G0(τ)) ≤ G0(τ0). In particular, if τ is square free, then G0(τ) is self-normalized in PSL2(ℝ). ©2000 American Mathematical Society.
Sat, 01 Jan 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036432000-01-01T00:00:00Z
- Ramanujan's modular equations and Atkin-Lehner involutionshttps://scholarbank.nus.edu.sg/handle/10635/104021Title: Ramanujan's modular equations and Atkin-Lehner involutions
Authors: Chan, H.H.; Lang, M.L.
Abstract: In this paper, we explain the existence of certain modular equations discovered by S. Ramanujan via function field theory. We will prove some of these modular equations and indicate how new equations analogous to those found in Ramanujan's notebooks can be constructed.
Thu, 01 Jan 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1040211998-01-01T00:00:00Z
- The signature of Γ+ 0(n)https://scholarbank.nus.edu.sg/handle/10635/104352Title: The signature of Γ+ 0(n)
Authors: Lang, M.-L.
Abstract: The signature of Γ+ 0(eN2), where e is square free, is completely determined if e=3 or N is odd or e∈Ψ, where Ψ is the set of all square free integers e∈N such that (i) if is odd, then admits no divisors of the form 8+3, (ii) if is even, then admits no divisors of the form 8+7.In particular, v2 (number of elliptic classes of period 2 of Γ+ 0(n)) is expressed as a linear combination of multiplicative functions. © 2001 Academic Press.
Sun, 01 Jul 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043522001-07-01T00:00:00Z
- Some modular functions associated to the Lie algebra E 8https://scholarbank.nus.edu.sg/handle/10635/104164Title: Some modular functions associated to the Lie algebra E 8
Authors: Chan, S.-P.; Lang, M.-L.; Lim, C.-H.
Tue, 01 Dec 1992 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041641992-12-01T00:00:00Z
- Extending π-systems to bases of root systemshttps://scholarbank.nus.edu.sg/handle/10635/103255Title: Extending π-systems to bases of root systems
Authors: Aslaksen, H.; Lang, M.L.
Abstract: Let R be an indecomposable root system. It is well known that any root is part of a basis B of R. But when can you extend a set, C, of two or more roots to a basis B of R? A π-system is a linearly independent set of roots such that if α and β are in C, then α - β is not a root. We will use results of Dynkin and Bourbaki to show that with two exceptions, A3Bn and A7E8, an indecomposable π-system whose Dynkin diagram is a subdiagram of the Dynkin diagrams of R can always be extended to a basis of R. © 2005 Published by Elsevier Inc.
Sun, 15 May 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032552005-05-15T00:00:00Z
- On Thompson's finiteness theoremhttps://scholarbank.nus.edu.sg/handle/10635/103848Title: On Thompson's finiteness theorem
Authors: Lang, M.L.
Abstract: In this article, we give an alternative (constructive) proof of Thompson's finiteness theorem. © 2004 Elsevier inc. All rights reserved.
Tue, 15 Feb 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1038482005-02-15T00:00:00Z
- Indecomposable Sylow 2-subgroups of simple groupshttps://scholarbank.nus.edu.sg/handle/10635/104576Title: Indecomposable Sylow 2-subgroups of simple groups
Authors: Harada, K.; Lang, M.L.
Abstract: Let S be a Sylow 2-subgroup of a finite simple group and let S = S 1 × S2 × Sk be the direct product and each component Si-, i = 1, 2,..., k is indecomposable. In this article, we prove that each Si is also a Sylow 2-subgroup of some simple group. © springer 2005.
Sat, 01 Jan 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1045762005-01-01T00:00:00Z
- A simple proof of the Markoff conjecture for prime powershttps://scholarbank.nus.edu.sg/handle/10635/102761Title: A simple proof of the Markoff conjecture for prime powers
Authors: Lang, M.L.; Tan, S.P.
Abstract: We give a simple and independent proof of the result of Jack Button and Paul Schmutz that the Markoff conjecture on the uniqueness of the Markoff triples (a, b, c) where a ≤ b ≤ c holds whenever c is a prime power. We also indicate some further directions for investigation. © 2007 Springer Science+Business Media B.V.
Mon, 01 Oct 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027612007-10-01T00:00:00Z
- Normalisers of subgroups of the modular grouphttps://scholarbank.nus.edu.sg/handle/10635/103640Title: Normalisers of subgroups of the modular group
Authors: Lang, M.L.
Abstract: Let G be a subgroup of finite index of the modular group Γ and let N(G) be the normaliser of G in PSL2(ℝ). In this article, we give an algorithm that determines N(G). © 2002 Elsevier Science (USA).
Fri, 01 Feb 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036402002-02-01T00:00:00Z
- Groups commensurable with the modular grouphttps://scholarbank.nus.edu.sg/handle/10635/103357Title: Groups commensurable with the modular group
Authors: Lang, M.L.
Abstract: Let A and B be maximal subgroups of PSL2(ℝ) that commensurable with PSL2(ℤ) (S and T are commensurable with each other if S ∩ T is of finite index in both S and T). In this article, we determine the index [A: A ∩ B] and the level of A ∩ B (Appendix A). © 2004 Elsevier Inc. All rights reserved.
Thu, 15 Apr 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033572004-04-15T00:00:00Z
- Independent generators for congruence subgroups of Hecke groupshttps://scholarbank.nus.edu.sg/handle/10635/103413Title: Independent generators for congruence subgroups of Hecke groups
Authors: Lang, M.-L.; Lim, C.-H.; Tan, S.-P.
Fri, 01 Dec 1995 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034131995-12-01T00:00:00Z
- On Rademacher's conjecture: Congruence subgroups of genus zero of the modular grouphttps://scholarbank.nus.edu.sg/handle/10635/103746Title: On Rademacher's conjecture: Congruence subgroups of genus zero of the modular group
Authors: Chua, K.S.; Lang, M.L.; Yang, Y.
Abstract: We list all genus zero congruence subgroups of PSL2(ℤ). There are altogether 132 of them (up to conjugation in PSL2(Zdbl;)). Geometrical invariants (genus, v2, v3, number of cusps, index in PSL2(ℤ)), fundamental polygons, Farey symbol, and independent generators of all such groups are determined. Generators of the function fields associated to such groups are also determined (http://www.math.nus.edu.sg/~matlml/). © 2004 Elsevier Inc. All rights reserved.
Thu, 01 Jul 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037462004-07-01T00:00:00Z
- Normalizers of the Congruence Subgroups of the Hecke Groups G4 and G6https://scholarbank.nus.edu.sg/handle/10635/103644Title: Normalizers of the Congruence Subgroups of the Hecke Groups G4 and G6
Authors: Lang, M.-L.
Abstract: Normalizers of Γ0(m)+w2 and Γ0(m)+w3 in PSL2(R) are determined. The determination of such normalizers enables us to determine the normalizers (in PSL2(R)) of the congruence subgroups G0 4(A) and G0 6(A) of the Hecke groups G4 and G6. © 2001 Academic Press.
Sat, 01 Sep 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036442001-09-01T00:00:00Z
- Normalizer of Γ1(m)https://scholarbank.nus.edu.sg/handle/10635/103641Title: Normalizer of Γ1(m)
Authors: Lang, M.-L.
Abstract: Let m∈N. Denote by N(Γ1(m)) the normalizer of Γ1(m) in PSL2(R). Then (i) N(Γ1(4))=N(Γ0(4))=Γ0(22)+; (ii) if m≠4, then N(Γ1(m))=Γ0(m)+. © 2001 Academic Press.
Mon, 01 Jan 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036412001-01-01T00:00:00Z
- The structure of the normalisers of the congruence subgroups of the Hecke group G5https://scholarbank.nus.edu.sg/handle/10635/104365Title: The structure of the normalisers of the congruence subgroups of the Hecke group G5
Authors: Lang, M.L.
Abstract: Let λ = 2cos (π/5) and let τ ∈ ℤ[λ]. Denote the normaliser of G0(τ) of the Hecke group G5 in PSL2(ℝ) by N(G0(τ)). Then N(G0(τ)) = G0(τ/h), where h is the largest divisor of 4 such that h 2 divides τ. Further, N(G0(τ))/G 0(τ) is either 1 (if h = 1), ℤ2 × ℤ2 (if h = 2) or ℤ4 × ℤ4 (if h = 4). © 2006 London Mathematical Society.
Thu, 01 Feb 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043652007-02-01T00:00:00Z