Search

Abstract: The main result in the paper states the following: Let π be a set of odd primes. Let the finite group be the product of a π-decomposable subgroup and a π-subgroup B. Then ; equivalently the group G possesses Hall π-subgroups. In this case is a Hall π-subgroup of G. This result extends previous results of Berkovich (1966), Rowley (1977), Arad and Chillag (1981) and Kazarin (1980) where stronger hypotheses on the factors A and B of the group G were being considered. The results under consideration in the paper provide in particular criteria for the existence of non-trivial soluble normal subgroups for a factorized group G. [Copyright &y& Elsevier]

Abstract: This is the second in a series of four papers on fixed point ratios in non-subspace actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either for all elements of prime order, or is one of a small number of known exceptions. In this paper we record a number of preliminary results and prove the main theorem in the case where the stabiliser is contained in a maximal non-subspace subgroup which lies in one of the Aschbacher families , where . [Copyright &y& Elsevier]

Abstract: This paper is devoted to the conjecture saying that, for any connected locally finite graph and any vertex-transitive group of automorphisms of , at least one of the following assertions holds: (1) There exists an imprimitivity system of on with finite (maybe one-element) blocks such that the stabilizer of a vertex of the factor graph in the induced group of automorphisms is finite. (2) The graph is hyperbolic (i.e., for some positive integer , the graph defined by and contains the regular tree of valency 3). Our approach to the conjecture consists in fixing a finite permutation group and considering the conjecture under the assumption that the stabilizer of a vertex of in induces on the neighborhood of the vertex a group permutation isomorphic to . In the paper we elaborate a method (the modified track method) which allows us to prove the conjecture for many groups . The paper consists of two parts. The present first part of the paper involves results on which the modified track method arguments are based, and a few first applications of the method. The second part is devoted to applications of the modified track method. [Copyright &y& Elsevier]

All groups considered in this paper will be finite. A 2-group is called quaternion-free if it has no section isomorphic to the quaternion group of order 8. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω1(P). Zhang [Proc. Amer. Math. Soc. 98 (4) (1986) 579] proved that if P is a Sylow p-subgroup of G, Ω1(P)⩽Z(P) and NG(Z(P)) is p-nilpotent, then G is p-nilpotent, i.e., G has a normal Hall p′-subgroup. Recently, Ballester-Bolinches and Guo [J. Algebra 228 (2000) 491] proved that if P is a Sylow 2-subgroup G, P is quaternion-free, Ω1(P∩G′)⩽Z(P) and NG(P) is 2-nilpotent, then G is 2-nilpotent. Bannuscher and Tiedt [Ann. Univ. Sci. Budapest 37 (1994) 9] proved that if p>2, P is a Sylow p-subgroup of G, &z.sfnc;Ω1(P∩Px)&z.sfnc;⩽pp-1 for all x∈G&z.drule;NG(P) and NG(P) is p-nilpotent, then G is p-nilpotent. The object of this paper is to improve and extend these results. [Copyright &y& Elsevier]

A subsemigroup S of a semigroup Q is a local left order in Q if, for every group H-class H of Q, S∩H is a left order in H in the sense of group theory. That is, every q∈H can be written as a♯b for some a,b∈S∩H, where a♯ denotes the group inverse of a in H. On the other hand, S is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as c♯d where c,d∈S and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. If one also insists that c and d can be chosen such that c R d in Q, then S is said to be a straight left order in Q.This paper investigates the close relation between local left orders and straight left orders in a semigroup Q and gives some quite general conditions for a left order S in Q to be straight. In the light of the connection between locality and straightness we give a complete description of straight left orders that improves upon that in our earlier paper. [Copyright &y& Elsevier]

Abstract: In this paper we use projectivity relative to kG-modules to define groups of relatively endotrivial modules, which are obtained by replacing the notion of projectivity with that of relative projectivity in the definition of ordinary endotrivial modules. To achieve this goal we develop the theory of projectivity relative to modules with respect to standard group operations such as induction, restriction and inflation. As a particular example, we show how these groups can generalise the Dade group. Finally, for finite groups having a cyclic Sylow p-subgroup, we determine all the different subcategories of relatively projective modules and, using the structure of the group of endotrivial modules described in Mazza and Thévenaz (2007) , the structure of all the different groups of relatively endotrivial modules. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study the structure of finite groups whose non-subnormal subgroups lie in some subclasses of the class of finite supersoluble groups. [Copyright &y& Elsevier]

Abstract: In this paper, we verify Dade''s invariant conjecture for Steinberg''s triality groups in the defining characteristic, i.e., in characteristic 2. Together with the results in [J. An, Dade''s conjecture for Steinberg triality groups in non-defining characteristics, Math. Z. 241 (2002) 445–469] and [J. An, F. Himstedt, S. Huang, Uno''s invariant conjecture for Steinberg''s triality groups in defining characteristic, in preparation], this completes the proof of Dade''s conjecture for Steinberg''s triality groups. Furthermore, we show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for in the defining characteristic, i.e., is good for the prime 2 in the sense of Isaacs, Malle and Navarro. [Copyright &y& Elsevier]

Abstract: A pro-p group G is a PF-group if it has central series of closed subgroups with trivial intersection satisfying and . In this paper, we prove that a finitely generated pro-p group G is a p-saturable group, in the sense of Lazard, if and only if it is a torsion free PF-group. Using this characterization, we study certain families of subgroups of p-saturable groups. For example, we prove that any normal subgroup of a p-saturable group contained in the Frattini is again p-saturable. [Copyright &y& Elsevier]

Abstract: This paper concerns the embedding problem for partial totally symmetric quasigroups. For all , it is shown that any partial totally symmetric quasigroup of order n can be embedded in a totally symmetric quasigroup of order v if v is even and , and this is the best possible such inequality. [Copyright &y& Elsevier]

Abstract: Let G be a finite group of characteristic p of one of the types symplectic group , Chevalley group , Suzuki group or Ree group . In this paper we show that if H is a Sylow p-subgroup of G then the restriction of χ to H has at least one linear constituent of multiplicity one for certain irreducible characters χ of G. [Copyright &y& Elsevier]

Abstract: Let N be a normal subgroup of a p-solvable group G and let M be a simple FN-module, where F is an algebraically closed field of characteristic p. Let Q be any p-subgroup of G. In this paper, we count the number of simple FG-modules lying over M and having Q as a vertex. [Copyright &y& Elsevier]

Abstract: In this paper we deal with equitable partitions of association schemes. We try to generalize a result in group theory and show examples that a generalization of a certain property conjectured for permutation groups does not hold for association schemes. [Copyright &y& Elsevier]

Abstract: Let be a regular uncountable cardinal and . The principle of Stationary Reflection in has been successful in settling problems of infinitary combinatorics in the case . For the principle is known to fail if is large enough. In this paper the principle is shown to fail for every . [Copyright &y& Elsevier]

In this paper, we study finite primitive permutation groups with a small suborbit. Based on the classification result of Quirin [Math. Z. 122 (1971) 267] and Wang [Comm. Algebra 20 (1992) 889], we first produce a precise list of primitive permutation groups with a suborbit of length 4. In particular, we show that there exist no examples of such groups with the point stabiliser of order 2436, clarifying an uncertain question (since 1970s). Then we analyse the orbital graphs of primitive permutation groups with a suborbit of length 3 or of length 4. We obtain a complete classification of vertex-primitive arc-transitive graphs of valency 3 and valency 4, and we prove that there exist no vertex-primitive half-arc-transitive graphs of valency less than 10. Finally, we construct vertex-primitive half-arc-transitive graphs of valency 2k for infinitely many integers k, with 14 as the smallest valency. [Copyright &y& Elsevier]

This paper studies the stable reduction of p-cyclic covers X→PK1 of the projective line over p-adic fields. So far, an algorithm to effectively determine the stable reduction of such covers is only known under additional hypothesis on the branch locus of the cover. Here, rather than restricting the type of cover, we consider the general case and obtain results on the structure of the special fiber Xk of the stable reduction of X. Special attention is payed to making all constructions effective. The central result is a formula computing the number of vanishing cycles on Xk. In particular, we give criteria for the special fiber of the stable reduction to be tree-like and for when X is a Mumford curve. Refining the analysis of vanishing cycles, we describe an algorithm that computes all the components of positive p-rank in the stable model. [Copyright &y& Elsevier]

Abstract: In this short note we show that we must replace the notion of connectedness used in the second author''s paper [M. Edmundo, Locally definable groups in o-minimal structures, J. Algebra 301 (2006) 194–223]. In particular, we show that with the current notion of connectedness some statements are false. [Copyright &y& Elsevier]