Note on normal subgroups of the modular group

On groups with uncountably many subgroups of finite index

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

In this paper, we shall deal with periodic groups, in which each element has a prime power order. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each [Formula: see text]-element of [Formula: see text] is of order [Formula: see text]. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each finite [Formula: see text]-subgroup of [Formula: see text] is of order [Formula: see text]. Here, [Formula: see text] denotes the set of all primes dividing the order of some element of [Formula: see text]. Our main results are the following four theorems. Theorem 1: Let [Formula: see text] be a finitely generated [Formula: see text]-group. Then [Formula: see text] has only a finite number of normal subgroups of finite index. Theorem 4: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a locally finite group. Theorem 7: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a finite group. Theorem 8: Let [Formula: see text] be a [Formula: see text]-group satisfying [Formula: see text]. Then [Formula: see text] is a locally finite group.

Generalizing results in [1], [3], [5], and [7] it is shown that the algebra generated by the quasi-regular representation λ of a discrete groupG onl2 (G/H) satisfies a standard polynomial identity if and only if λ (G) contains an abelian normal subgroup of finite index. An analogon for topological groups is also proved.

Let \(\mathcal{N}\) be a nonempty collection of normal subgroups of finite index of a group G and assume that \(\mathcal{N}\) is filtered from below, i.e., \(\mathcal{N}\) satisfies the following condition: $$\mbox{whenever }N_1,N_2\in \mathcal{N},\mbox{ there exists }N\in \mathcal{N}\mbox{ such that }N\le N_1\cap N_2.$$ Then one can make G into a topological group by considering \(\mathcal{N}\) as a fundamental system of neighborhoods of the identity element 1 of G. We refer to the corresponding topology on G as a profinite topology. If every quotient G/N\((N\in \mathcal{N})\) belongs to a certain class \(\mathcal{C}\), we say more specifically that the topology above is a pro - \(\mathcal{C}\)topology.

Profinite completions and isomorphic finite quotients

The uniqueness of the translation-invariant extreme Gibbs measure for the antiferromagnetic Potts model with an external field and the existence of an uncountable number of extreme Gibbs measures for the Ising model with an external field on the Cayley tree are proved. The classes of normal subgroups of finite index of the Cayley tree group representation are constructed. The periodic extreme Gibbs measures, which are invariant with respect to subgroups of index 2, are constructed for the Ising model with zero external field. From these measures, the existence of an uncountable number of nonperiodic extreme Gibbs measures for the antiferromagnatic Ising model follows.

Let $G$ be a finite $p$-group of order $p^{m}=p^{2n+e}$, with $n$ a non-negative integer, $p$ a prime number and $e=0$ or $1$, and let $r(G)$ be the number of conjugacy classes of elements of $G$. Then the following equality, due to P. Hall, holds ([4], p. 549) : $$r(G)=(p^{2}-1)n+p^{e}+k(p^{2}-1)(p-1) ,$$ For some non-negative integer $k$. In this paper, we obtain new properties relative to $r(G)$ by the analysis of the number $r_{G}(gN)$ of conjugacy classes of elements of $G$ that intersect the coset $gN$, where $N$ is a normal subgroup of $G$ and $g$ any element of $G$. It contains a number of equations and congruences relating $r(G)$ to other invariants of $G$. In particular, our results improve the above equality of P. Hall, when $G$ has maximal nilpotent class or $n\leq p+1$. Examples are given, which make our improvements evident.

We study the class number one problem for a one parameter family of real quadratic fields $\mathbb{Q}(\sqrt{d})$, where $d=100m^2+ 28m+2$ and $m$ is a positive integer. We show that there is no such field with class number one.

This paper presents a list of all finite groups having exactly six and seven conjugate classes and an outline of the background necessary for the proof, and gives, in particular, two results which may be of independent interest. In 1903 E. Landau (8) proved, by induction, that for each the equation*has only finitely many solutions over the positive integers.

Let G be a compact Hausdorff group and n a positive integer. It is proved that all subnormal subgroups of G of index dividing n are open if and only if there are only finitely many such subgroups, and that all subgroups of finite index in G are open if and only if there are only countably many such subgroups.

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let α be an arbitrary real root of a quadratic equation with integer coefficients; say, α = \( \sqrt 2 \). Given any rational number 0 < x < 1 (say, x = 1/2) and any positive integer n, we count the number of elements of the sequence α, 2α, 3α, …, nα modulo 1 that fall into the subinterval [0, x]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected number” nx from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ n ≤ N. Depending on α and x, we may need an extra additive correction of constant times logarithm of N; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm of N. If N is large, the distribution of this renormalized counting number, as n runs in 1 ≤ n ≤ N, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as N tends to infinity. This is the main result of the paper (see Theorem 1.1). The proof is rather complicated and long; it has many interesting detours and byproducts. For example, the exact determination of the key constant factors (in the additive and multiplicative norming), which depend on α and x, requires surprisingly deep algebraic tools such as Dedeking sums, the class number of quadratic fields, and generalized class number formulas. The crucial property of a quadratic irrational is the periodicity of its continued fraction. Periodicity means self-similarity, which leads us to Markov chains: our basic probabilistic tool to prove the central limit theorem. We also use a lot of Fourier analysis. Finally, I just mention one byproduct of this research: we solve an old problem of Hardy and Littlewood on diophantine sums.

In a recent paper, Guerzhoy obtained formulas for certain class numbers as p-adic limits of traces of singular moduli. Using earlier work by Bruinier and the second author, we derive a more precise form of these results using results of Zagier. Specifically, if −d < −4 is a fundamental discriminant and n is a positive integer, then Tr(pd) ≡ 24 p− 1 · ( 1− (−d p )) ·H(−d) (mod p) provided that p ∈ {2, 3} and (−d p ) = −1, or p ∈ {5, 7, 13} and (−d p ) 6= 1.

Let K = Q ( θ ) \mathcal {K} = \mathcal {Q}(\theta ) be the algebraic number field formed by adjoining θ \theta to the rationals Q \mathcal {Q} . Let R and h be, respectively, the regulator and class number of K \mathcal {K} . Shanks has described a method of evaluating R for Q ( D ) \mathcal {Q}(\sqrt D ) , where D is a positive integer. His technique improved the speed of the usual continued fraction algorithm for finding R by allowing one to proceed almost directly from the n th to the m th step, where m is approximately 2 n , in the continued fraction expansion of D \sqrt D . This paper shows how Shanks’ idea can be extended to the Voronoi algorithm, which is used to find R in cubic fields of negative discriminant. It also discusses at length an algorithm for finding R and h for pure cubic fields Q ( D 3 ) \mathcal {Q}(\sqrt [3]{D}) , D an integer. Under a certain generalized Riemann Hypothesis the ideas developed here will provide a new method which will find R and h in O ( D 2 / 5 + ε ) O({D^{2/5 + \varepsilon }}) operations. When h is small, this is an improvement over the O ( D / h ) O(D/h) operations required by Voronoi’s algorithm to find R . For example, with D = 200171999 D = 200171999 , it required only 5 minutes for an AMDAHL 470/ V 7 computer to find that R = 518594546.969083 R = 518594546.969083 and h = 1 h = 1 . This same calculation would require about 8 days of computer time if it used only the standard Voronoi algorithm.

We develop a theory connecting the following three areas: (a) the mean field equation (MFE) 4u + eu = ρ δ0, ρ ∈ R>0 on flat tori Eτ = C/(Z + Zτ), (b) the classical Lame equations and (c) modular forms. A major theme in part I is a classification of developing maps f attached to solutions u of the mean field equation according to the type of transformation laws (or monodromy) with respect to Λ satisfied by f . We are especially interested in the case when the parameter ρ in the mean field equation is an integer multiple of 4π. In the case when ρ = 4π(2n + 1) for a non-negative integer n, we prove that the number of solutions is n + 1 except for a finite number of conformal isomorphism classes of flat tori, and we give a family of polynomials which characterizes the developing maps for solutions of mean field equations through the configuration of their zeros and poles. Modular forms appear naturally already in the simplest situation when ρ = 4π In the case when ρ = 8πn for a positive integer n, the solvability of the MFE depends on the moduli of the flat tori Eτ and leads naturally to a hyperelliptic curve Xn = Xn(τ) arising from the Hermite-Halphen ansatz solutions of Lame’s differential equation d2w dz2 − (n(n + 1)℘(z; Λτ) + B)w = 0. We analyse the curve Xn from both the analytic and the algebraic perspective, including its local coordinate near the point at infinity, which turns out to be a smooth point of Xn. We also specify the role of the branch points of the hyperelliptic projection Xn → P1 when the parameter ρ varies in a neighborhood of ρ = 8πn. In part II, we study a “premodular form” Zn(σ; τ), a real-analytic function in two variables associated to Xn(τ), which has many symmetries and also the property that the τ-coordinates of zeros of Zn(σ; τ) correspond exactly to those flat tori where the MFE with parameter ρ = 8πn has a solution.