The Levi classes generated by nilpotent groups

Given an arbitrary class M of groups, denote by L(M) the class of all groups G in which the normal closure of every element belongs to M. Consider the quasivariety _q F _p generated by the relatively free group in the class of nilpotent groups of length at most 2 with the commutant of exponent p (where p is an odd prime). We describe the Levi class that is generated by qF _p.     

Linear Bounds of Degrees of Alternating Quotients of the (3, q,r) Triangle Groups

Abstract

In Everitt [Everitt, B. J. (2000). Alternating quotients of Fuchsian groups. J. Algebra 223: 457–476], it was shown, in particular, that each Fuchsian triangle group Δ(p, q, r) has among its homomorphic images all but finitely many of the alternating groups. Treating p, q as fixed, the methods of Everitt (2000) give a quadratic function N(r) of r such that A _n is an image of Δ(p, q, r) for every integer n ≥ N(r). We conjecture that there is a linear function of r with this property. In this paper, we will show that the conjecture holds for the Fuchsian triangle groups Δ(3, q, r).     

A new characterization ofA p wherep andp − 2 are primes

Based on the prime graph of a finite simple group, its order is the product of its order components (see [4]). It is known that Suzuki-Ree groups [6],PSL ₂(q) [8] andE ₈(q) [7] are uniquely determined by their order components. In this paper we prove that the simple groupsA _p are also uniquely determined by their order components, wherep andp − 2 are primes.     

A reduction theorem for the McKay conjecture

The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL₂(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2 ^e or q=3 ^e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J ₁, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.     

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.     

Contributo allo studio del problema di hughes sui gruppi

Sunto. Hughes ha avanzato la seguente congettura: se G è un gruppo, p un numero primo, e H_p il sottogruppo generato dagli elementi di G che non hanno periodo p, si presenta uno dei tre casi seguenti: H_p=1; H_p=G; [G: H_p]=p. Vari autori hanno provato l'esattezza di tale congettura per larghe classi di gruppi. In questa Nota si dimostra che tale congettura è esatta per i p-gruppi finiti in cui ogni sottogruppo generato da tre elementi, due dei quali di periodo p, abbia classe ≤p; essa è quindi esatta in particolare per i p-gruppi di classe ≤p.

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Summary Hughes advanced the following conjecture: “let G be a group, p a prime and H_p the subgroup generated by elements of G having order p; then H_p=1 or H_p=G or [G: H_p]=p„. Several authors have proved this conjecture for various classes of groups. The Author proves Hughes conjecture for finite p-groups such that every subgroup {x ₁ , x _{2, y}} generated by two elements x ₁ , x ₂ having order p, and by a thirth element y, has class ≤p; in particular, the conjecture is proved for finite p-groups having class ≤p.

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A Enrico Bompiani in occasione del suo Giubileo scientifico.     

Semisimple metacyclic group algebras

Given a group G of order p ₁ p ₂, where p ₁, p ₂ are primes, and \mathbbF_q\mathbb{F}_{q}, a finite field of order q coprime to p ₁ p ₂, the object of this paper is to compute a complete set of primitive central idempotents of the semisimple group algebra \mathbbF_q[G]\mathbb{F}_{q}[G]. As a consequence, we obtain the structure of \mathbbF_q[G]\mathbb{F}_{q}[G] and its group of automorphisms.     

Brauer characters with cyclotomic field of values

It has been shown in an earlier paper [G. Navarro, Pham Huu Tiep, Rational Brauer characters, Math. Ann. 335 (2006) 675-686] that, for any odd prime p, every finite group of even order has a non-trivial rational-valued irreducible p-Brauer character. For p=2 this statement is no longer true. In this paper we determine the possible non-abelian composition factors of finite groups without non-trivial rational-valued irreducible 2-Brauer characters. We also prove that, if p≠q are primes, then any finite group of order divisible by q has a non-trivial irreducible p-Brauer character with values in the cyclotomic field Q(exp(2πi/q)).     

Weakly group-theoretical and solvable fusion categories

We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups – weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group-theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq², where p,q,r are distinct primes.     

The existence of J 3 and its embeddings in E 6

We determine the embeddings of the third sporadic group J ₃ of Janko in simple Chevalley groups of type E ₆ over finite and algebraically closed fields. As a corollary we obtain a short elegant existence proof of J ₃. This is of interest as J ₃ is one of the few sporadic groups not contained in the Monster, so its existence cannot be verified within that group. Previous existence proofs were highly computational; cf. [4] and [6].To Jacques Tits on his sixtieth birthdayPartially supported by NSF DMS-8721480 and NSA MDA90-88-H-2032.     

Graphs on which a group of order <Emphasis Type="Italic">pq</Emphasis> acts edge-transitively

Let F be a finite simple undirected graph with no isolated vertices. Let p, q be prime numbers with p≥q. We complete the classification of the graphs on which a group of order pq acts edge-transitively. The results are the following. If Aut（Г） contains a subgroup G of order pq that acts edge-transitively on F, then F is one of the following graphs： （1） pK1,1; （2） pqK1,1; （3） pgq,1; （4） qKp,1 （p 〉 q）; （5） pCq （q 〉 2）; （6） qCp （p 〉 q）; （7） Cp （p 〉 q = 2）; （8） Cpq; （9） （Zp, C） whereC={±r^μ ｜μ∈Zq} withq〉2, q｜（p-1） and r≠1≡r^q （modp）; （10） Kp,1 （p 〉 q）; （11） a double Cayley graph B（G,C） with C = {1-r^μ ｜ μ ∈ Zq} and r≠1≡r^q （modp）; （12） Kpq,1;or （13） Kp,q.     

Characterization of 3 D 4(q)

The author defined the concept order components in [2] and gave a new characterization of sporadic simple groups by their order components in [7]. Afterwards the following groups were characterized by the author: G₂(q), q = 0 (mod 3)^[8]; E₈(q)^[9]; Suzuki-Ree groups^[10]; PSL₂(q)^[11]. Here the author will continue such kind of characterization and prove that:Theorem 1. Let G be a finite group, M = ³D₄(q). If G and M has the same order components, then G M.And the following theorems follows from Theorem 1.Theorem 2. (Thompsons Conjecture) Let G be a finite group, Z(G) = 1,M = ³D₄(q). If N(G) = N(M), then G M. (ref. [6])Theorem 3. (Wujie Shi) Let G be a finite group, M = ³D₄(q). If|G| = |M|, _e(G) = _e(M), then G M. (ref. [15])All notations are the same as in [2]. According to the classification theorem of finite simple groups, [12] and [13], we can list the order components of finite simple groups with nonconnected prime graphs in Tables 1-4 (ref. [5]).American Mathematics Society Classification 20D05 20D60The author is indebted to Fred and Barbara Kort Sino-Israel Postdoctoral Programme for supporting my post-doctoral position (1999.10-2000.10) at Bar-Ilan University, also to Emmy Noether Mathematics Institute and NSFC for partially financial support.     

A tensor product of the Steinberg module and a certain simple kG(pr)-module

Let G be a connected, semisimple, and simply connected algebraic group defined and split over the finite field of order p, and let G(q) be the corresponding finite Chevalley or twisted group, where q = p^r. Recently, Anwar determines the direct sum decomposition of the tensor product of the rth Steinberg module and a simple G-module with a (p,r)-minuscule highest weight λ. In this paper, we determine that of the tensor product regarded as a module for G(q) under some weak assumptions for λ.     

The inverse Galois problem over formal power series fields

Consider a valuation ringR of a discrete Henselian field and a positive integerr. LetF be the quotient field of the ringR[[X ₁, …,X _r ]]. We prove that every finite group occurs as a Galois group overF. In particular, ifK ₀ is an arbitrary field andr≥2, then every finite group occurs as a Galois group overK ₀((X ₁, …,X _r )). The work on this paper started when the author was an organizer of a research group on the Arithmetic of Fields in the Institute for Advanced Studies at the Hebrew Univesity of Jerusalem in 1991–92. It was partially supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

nX-complementary generations of the Fischer group Fi 23

Let G be a finite group andnX a conjugacy class of elements of order n inG. G is callednX—complementary generated if, for everyx ∈ G?{1}, there is ay ∈nX such thatG = 〈hx,y〉. In [27] the question of finding all positive integersn such that a given non-abelian finite simple groupG isnX—-complementary generated was posed. In this paper we answer this question for the sporadic groupFi ₂₃. In fact, we prove that the Fischer groupFi ₂₃ isnX-complementary generated if and only ifn < 12 orn ∈ {7, 8, 10, 11} ornX ∈ {6N, 6O, 9D, 9E, 12C, 12D, …,12O}.     

Genus 2 Belyĭ surfaces with a unicellular uniform dessin

A bicoloured graph embedded in a compact oriented surface and dividing it into a union of simply connected components (faces) is known as a dessin d’enfant. It is well known that such a graph determines a complex structure on the underlying topological surface, but a given compact Riemann surface may correspond to different dessins. In this paper we deal with all unicellular (one-faced) uniform dessins of genus 2 and their underlying Riemann surfaces. A dessin is called uniform if white vertices, black vertices and faces have constant degree, say p, q and r respectively. A uniform dessin d’enfant of type (p, q, r) on a given surface S corresponds to the inclusion of the torsion-free Fuchsian group K uniformizing S inside a triangle group Δ(p, q, r). Hence the existence of different uniform dessins on S is related to the possible inclusion of K in different triangle groups. The main result of the paper states that two unicellular uniform dessins belonging to the same genus 2 surface must necessarily be isomorphic or obtained by renormalisation. The problem is approached through the study of the face-centers of the dessins. The displacement of such a point by the elements of K must belong to a prescribed discrete set of (hyperbolic) distances determined by the signature (p, q, r). Therefore looking for face-centers amounts to finding points correctly displaced by every element of K.     

Graded Nilpotent Algebras and Eggert's Conjecture

A group G is (l,m,n)-generated if it is a quotient group of the triangle group T(l,m,n) = (x,y,z|x ^l= y ^m= z ⁿ= xyz= 1). In [8] the problem is posed to find all possible (l,m,n)-generations for the non-abelian finite simple groups. In this paper we partially answer this question for the Janko group J ₃. We find all (2, 3, t)-generations as well as (2, 2,2,p)-generations, p a prime, for J ₃     

Geometries with dual affine planes and symplectic quadrics

Farmer and Hale [3] prove that every copolar space fully embedded in a finite projective space PG(n, q), with q>, is the copolar space arising from a symplectic polarity. We show that this result is still valid in arbitrary projective spaces; this provides a different and shorter proof of [3] in the finite case.     

Extensions of group representations from a normal subgroup

We characterize principal nilpotent ideals in finite abelian group algebra F_q [G] (F_q a finite field of p^s elements, p prime), with greatest possible dimension. We prove also that the-se ideals are all isomorphic.     

On PSL(2,q) as a totally irregular collineation group

Non-abelian simple totally irregular collineation groups containing an involutorial perspectivity have been classified by the authors in a recent paper. They are PSL(2,q), PSL(3,q), PSU(3,q), Sz(q), the alternating group on 7 letters, and the second Janko sporadic simple group. In this article, we study PSL(2,q),q congruent to 1 modulo 4, as a collineation group containing an involutory homology.C. Y. Ho was partially supported by a NSA grant.