Mong Lung Lang

RAMANUJAN'S MODULAR EQUATIONS AND ATKIN-LEHNER INVOLUTIONS

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.

Congruence Subgroups Associated to the Monster

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 PSL 2(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 separatetable (see www.rnath.nus.edu.sg/e-matlml/).

A simple proof of the Markoff conjecture for prime powers

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.

Groups commensurable with the modular group

Abstract Let A and B be maximal subgroups of PSL 2 ( R ) that commensurable with PSL 2 ( 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).

Finite quotients of the automorphism group of a free group

The automorphism group AutFn of a free group Fn of rank n acts on the product of n copies of a group G by substituting n elements of G into the words defining an automorphism of the free group. This gives rise to an antihomomorphism from AutFnto a permutation group. We determine this antihomomorphic image of AutFn when G is the semidirect product Zp x Zq

Sylow 2-Subgroups of Finite Simple Groups

To one’s surprise, it was only until the late of the 19th century that a mathematician announced the classification of all groups of order 12. Unfortunately there was an error. Three years later, in 1899, Cayley showed it correctly. Namely, there are five nonisomorphic groups of order 12. One hundred years is long enough for mathematicians to make a quantum leap, since in the year 2000, Besche, Eick, and O’Brien [BEO] determined all isomorphism classes of groups of order ≤ 2000. Among them, there are exactly 49,487,365,422 groups of order 1024 = 210. All others count 423,164,062 in number. In other words, 99.16% of all groups of order ≤ 2000 are of just one order 210. (If we add groups of order 512, 128, etc., the ratio will be only a little greater for 2-groups.) Asymptotically perhaps: Almost all finite groups are 2-groups.