Principal Congruence Subgroups of Hecke Groups H（√q）

Using the notion of quadratic reciprocity, we discuss the principal congruence subgroups of the Hecke groups H（√q）,q 〉 5 prime number.     

SOME SUBGROUPS OF THE EXTENDED HECKE GROUPS (λ_q)

The authors consider the extended Hecke groups H(λq) generated by T(z) = -1 / z, S(z) = -1/(z + λq,) and R(z) = 1 / z with A, = 2 cos(π/q) for q≥3 an integer. In this paper, the even subgroup He(λq), the second commutator subgroup H (λq) and the principal congruence subgroups Hp(λq) of the extended Hecke groups H(λq) are studied. Also, relations between them are given.     

SOME SUBGROUPS OF THE EXTENDED HECKE GROUPS (H)(λq)

The authors consider the extended Hecke groups H(λq) generated by T(z) =-1 / z, S(z) = -1/(z+λq) and R(z) = 1 / -z with λq = 2cos(π/q) for q ≥ 3 an integer. Inthis paper, the even subgroup He(λq), the second commutator subgroup H"(λq) and theprincipal congruence subgroups Hp(λq) of the extended Hecke groups H(λq) are studied.Also, relations between them are given.     

REPRESENTATIONS OF AFFINE HECKEALGE BRAS OF TYPE G2

Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affne Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representations of Hq by using based rings of two-sided cells of an affne Weyl group W of type G2. We shall give the classification of irreducible representations of Hq. We also remark that a calculation in [11] actually shows that Theorem 2 in [1] needs a modification, a fact is known to Grojnowski and Tanisaki long time ago. In this paper we also show an interesting relation between Hq and an Hecke algebra corresponding to a certain Coxeter group. Apparently the idea in this paper works for all affne Weyl groups, but that is the theme of another paper.     

On Nearly SS-Embedded Subgroups of Finite Groups

Let H be a subgroup of a finite group G. H is nearly SS-embedded in G if there exists an S-quasinormal subgroup K of G, such that HK is S-quasinormal in G and H ∩ K ≤ HseG, where HseG is the subgroup of H, generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, the authors investigate the influence of nearly SS-embedded subgroups on the structure of finite groups.     

利用GAP研究一般线性群的抛物子群

A typical example for the algebraic groups is the general linear groups G = GL(n,F), we have studied the structure of such groups and paid special attention to its important substructures, namely the Parabolic subgroups.For a given G we computed all the Parabolic subgroups and determined their number, depending on the fact that any finite group has a composition series and the composition factors of a composition series are simple groups which are completely classified, we report here some investigations on the computed Parabolic subgroups. This has been done with the utility of GAP.     

A Sufficient Condition for Mi-groups

In this note, we give a sufficient condition for Mi-group. In particular, we show that if a finite group G is the semidirect product of two subgroups with coprime orders, in which one is a Sylow tower group and its Sylow subgroups are all abelian, and the other is an Mi-group and all of its proper subgroups are also Mi-groups, then G is an Mi-group.     

Mi群的一个充分条件

In this note, we give a sufficient condition for Mi-group. In particular, we show that if a finite group G is the semidirect product of two subgroups with coprime orders, in which one is a Sylow tower group and its Sylow subgroups are all abelian, and the other is an Mi-group and all of its proper subgroups are also Mi-groups, then G is an Mi-group.     

Finite p-Groups in Which the Number of Subgroups of Possible Order is Less Than or Equal to $p^3 $

In this paper, groups of order pn in which the number of subgroups of possible order is less than or equal to p3 are classified. It turns out that if p 2, n ≥ 5, then the classification of groups of order pn in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order pn with a cyclic subgroup of index p2 are the same.     

Equicentralizer Subgroups of Sporadic Simple Groups

A subgroup H of a group C is called an equicentralizer subgroup of G if thecentralizers of its nonidentity elements are all equal. We discuss the equicentrali-zer subgroups of sporadic simple groups. Using the orders of centralizers of theelements and the structure of the maximal subgroups, we prove that the equicent-     

Principal congruence subgroups of the Hecke groups and related results

In this paper, first, we determine the quotient groups of the Hecke groups H(λ _q ), where q ≥ 7 is prime, by their principal congruence subgroups H _p (λ _q ) oflevel p, where p is also prime. We deal with the case of q = 7 separately, because of its close relation with the Hurwitz groups. Then, using the obtained results, we find the principal congruence subgroups of the extended Hecke groups $ \overline H $ \overline H (λ _q ) for q ≥ 5 prime. Finally, we show that some of the quotient groups of the Hecke group H(λ _q ) and the extended Hecke group $ \overline H $ \overline H (λ _q ), q ≥ 5 prime, by their principal congruence subgroups H _p (λ _q ) are M*-groups.     

Parity patterns associated with lifts of Hecke groups

Let q be an odd prime, m a positive integer, and let Γ _m (q) be the group generated by two elements x and y subject to the relations x ^2m =y ^qm =1 and x ²=y ^q ; that is, Γ _m (q) is the free product of two cyclic groups of orders 2m respectively qm, amalgamated along their subgroups of order m. Our main result determines the parity behaviour of the generalized subgroup numbers of Γ _m (q) which were defined in Müller (Adv. Math. 153:118–154, 2000), and which count all the homomorphisms of index n subgroups of Γ _m (q) into a given finite group H, in the case when gcd (m,| H |)=1. This computation depends upon the solution of three counting problems in the Hecke group ℋ(q)=C ₂*C _q : (i) determination of the parity of the subgroup numbers of ℋ(q); (ii) determination of the parity of the number of index n subgroups of ℋ(q) which are isomorphic to a free product of copies of C ₂ and of C _∞; (iii) determination of the parity of the number of index n subgroups in ℋ(q) which are isomorphic to a free product of copies of C _q . The first problem has already been solved in Müller (Groups: Topological, Combinatorial and Arithmetic Aspects, LMS Lecture Notes Series, vol. 311, pp. 327–374, Cambridge University Press, Cambridge, 2004). The bulk of our paper deals with the solution of Problems (ii) and (iii). Research of C. Krattenthaler partially supported by the Austrian Science Foundation FWF, grant S9607-N13, in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

Branch groups

The class of branch groups is defined (both in the abstract and in the profinite category). The relationship of this class with the class of extremal groups is established. Properties of the branch groups are investigated. Applications of the congruence property to the theory of profinite branch groups are indicated. The weak maximality of parabolic subgroups in branch groups is proved. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 852–858, June, 2000.     

Congruences Concerning Values of the Modular Functions <Emphasis Type="Italic">j</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The j-function j(z) = q⁻¹+ 744 + 196884q + ⋅s plays an important role in many problems. In [7], Zagier, presented an interesting series of functions obtained from the j-function: j_m(ζ) = (j(ζ) – 744)∨T₀(m), where T₀(m) is the usual m′th normalized weight 0 Hecke operator. In [3], Bruinier et al. show how this series of functions can be used to describe all meromorphic modular forms on SL₂(ℤ). In this note we use these functions and basic notions about modular forms to determine previously unidentified congruence relations between the coefficients of Eisenstein series and the j-function. 2000 Mathematics Subject Classification: Primary–11B50, 11F03, 11F30 The author thanks the National Science Foundation for their generous support.     

Integral mean values of modular L-functions

We consider the mean squares of L-functions associated to modular forms with respect to Hecke congruence subgroups, expressing the mean value as an inner product. This avoids the discussion of generalized additive divisor problems. As applications, we obtain asymptotic formulas for both weighted and unweighted mean squares.     

The normal subgroup structure of the extended Hecke groups

We consider the extended Hecke groups generated by T(z) = −1/z, S(z) = −1/(z + λ) and R(z) = 1/z with λ ≥ 2. In this paper, firstly, we study the fundamental region of the extended Hecke groups . Then, we determine the abstract group structure of the commutator subgroups , the even subgroup , and the power subgroups of the extended Hecke groups . Also, finally, we give some relations between them.     

Topological equivalence of Haar measures on compact groups

The existence of homeomorphisms establishign an isometry of normalized Haar measures on (metrizable) compact groups is studied. In the case of 0-dimensional groups, a complete answer is given in terms of the indices of open normal subgroups. For example, for the countable powers of the groups ℤ/(m) and ℤ/(n), the answer is affirmative if and only ifm andn have the same prime divisors. A certain class of extensions of 0-dimensional groups is also studied. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 188–194, August, 2000.     

Normal Subgroups of Hecke Groups <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">λ</Emphasis>)

Let λ ≥ 2 and let H(λ) be the Hecke group associated to λ. Also let H(λ)\U be the Riemann surface associated to the Hecke group H(λ). In this article, we study the even subgroup H _e (λ) and the power subgroups H ^m (λ) of the Hecke groups H(λ). We also study some genus 0 normal subgroups of finite index of H(λ). Finally, we discuss free normal subgroups of H(λ).     

Normalizers of subsystem subgroups in finite groups of Lie type

Finite groups of Lie type form the greater part of known finite simple groups. An important class of subgroups of finite groups of Lie type are so-called reductive subgroups of maximal rank. These arise naturally as Levi factors of parabolic groups and as centralizers of semisimple elements, and also as subgroups with maximal tori. Moreover, reductive groups of maximal rank play an important part in inductive studies of subgroup structure of finite groups of Lie type. Yet a number of vital questions dealing in the internal structure of such subgroups are still not settled. In particular, we know which quasisimple groups may appear as central multipliers in the semisimple part of any reductive group of maximal rank, but we do not know how normalizers of those quasisimple groups are structured. The present paper is devoted to tackling this problem. Supported by RFBR (grant No. 05-01-00797) and by SB RAS (Young Researchers Support grant No. 29 and Integration project No. 2006.1.2). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 3–30, January–February, 2008.