On p-nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S4-free and every minimal subgroup of P n GN is c-supplemented in NG(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

Cyclic normal subgroups of generalized Baumslag–Solitar groups

The centers and cyclic radicals of generalized Baumslag–Solitar groups (GBS-groups) are studied. Criteria for their non-triviality are found and an algorithm to compute the center and cyclic radical is given. In addition, for each unimodular GBS-group, a GBS-cover, i.e., a GBS-group in whose underlying graph the weights around each vertex are constant in absolute value, is constructed. These groups are then characterized by a minimality property.     

Open subgroups of locally compact Kac–Moody groups

Let $G$ be a complete Kac–Moody group over a finite field. It is known that $G$ possesses a BN-pair structure, all of whose parabolic subgroups are open in $G$ . We show that, conversely, every open subgroup of $G$ is contained with finite index in some parabolic subgroup; moreover there are only finitely many such parabolic subgroups. The proof uses some new results on parabolic closures in Coxeter groups. In particular, we give conditions ensuring that the parabolic closure of the product of two elements in a Coxeter group contains the respective parabolic closures of those elements.     

p-Hypercyclically embedding and Π-property of subgroups of finite groups

Let G be a finite group, E a normal subgroup of G and p a fixed prime. We say that E is p-hypercyclically embedded in G if every p-G-chief factor of E is cyclic. A subgroup H of G is said to satisfy Π-property in G if |G∕K:N_G∕K((H∩L)K∕K)| is a π((H∩L)K∕K)-number for any chief factor L∕K in G; we say that H has Π*-property in G if H∩O^π(H)(G) has Π-property in G. In this paper, we prove that E is p-hypercyclically embedded in G if and only if some classes of p-subgroups of E have Π*-property in G. Some recent results are extended.     

A chiral Borel–Weil–Bott theorem

We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. This is applied to (1) finding the character of G-integrable irreducible highest weight modules over the affine Lie algebra at the critical level, and (2) computing a certain elliptic genus of the flag manifold. The main tool is a result that interprets the Drinfeld–Sokolov reduction as a derived functor.     

On δ-permutability of maximal subgroups of Sylow subgroups of finite groups

In this paper, we get the main theorem: Let p be a prime dividing the order of G and , where and H is p ^′-Hall subgroup of G. Let δ be a complete set of Sylow subgroups of H. If G satisfies the following conditions: i) is a p-group; ii) for any maximal M of P, M is δ-permutable in H, then G is p-nilpotent. Some known results are generalized. Received: 12 September 2007, Revised: 28 February 2008     

Commutator subgroups of the power subgroups of?some Hecke groups

Let q??3 be a prime and let H(?? _q ) be the Hecke group associated to q. Let m be a positive integer and H ^m (?? _q ) be the mth power subgroup of H(?? _q ). In this work, we study the commutator subgroups of the power subgroups H ^m (?? _q ) of H(?? _q ). Then, we give the derived series for all triangle groups of the form (0;2,q,n) for n a positive integer, since there is a nice connection between the signatures of the subgroups we studied and the signatures of these derived series.     

Finite groups with some maximal subgroups of Sylow subgroups ℳ-supplemented

A subgroup H of a group G is said to be ?-supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H. In this paper, we obtain the following statement: Let ? be a saturated formation containing all supersolvable groups and H be a normal subgroup of G such that G/H ε ?. Suppose that every maximal subgroup of a noncyclic Sylow subgroup of F*(H), having no supersolvable supplement in G, is ?-supplemented in G. Then G ε ?.     

On Mordell–Weil groups of isotrivial abelian varieties over function fields

We show that the Mordell–Weil rank of an isotrivial abelian variety with cyclic holonomy depends only on the fundamental group of the complement to the discriminant, provided the discriminant has singularities in CM class introduced here. This class of singularities includes all unibranched plane curves singularities. As a corollary, we describe a family of simple Jacobians over the field of rational functions in two variables for which the Mordell–Weil rank is arbitrarily large.     

Separability of the subgroups of residually nilpotent groups in the class of finite π-groups

Siberian Mathematical Journal - Given a nonempty set π of primes, call a nilpotent group π-bounded whenever it has a central series whose every factor F is such that: In every quotient...     

A Borel–Weil–Bott type theorem for group completions

We prove a character formula of cohomologies of line bundles on the wonderful completion of an adjoint semisimple algebraic group in the sense of De Concini and Procesi (in our case, we need the general construction of De Concini and Springer).     

On K-ℙ-subnormal subgroups of finite groups

A subgroup H of a group G is said to be K-?-subnormal in G if H can be joined to the group by a chain of subgroups each of which is either normal in the next subgroup or of prime index in it. Properties of K-?-subnormal subgroups are obtained. A class of finite groups whose Sylow p-subgroups are K-?-subnormal in G for every p in a given set of primes is studied. Some products of K-?-subnormal subgroups are investigated.     

Finite groups with systems of Σ-embedded subgroups

In recent years,a series of papers about cover-avoiding property of subgroups appeared and all the studies were connected with chief factors of a finite group.However,about the cover-avoiding property of subgroups for non-chief factor,there is no study up to now.The purpose of this paper is to build the theory.Let A be a subgroup of a finite group G and Σ:G0≤G1≤…≤Gn some subgroup series of G.Suppose that for each pair(K,H) such that K is a maximal subgroup of H and G i 1 K < H G i for some i,either A ∩ H = ...     

On the products of ℙ-subnormal subgroups of finite groups

A subgroup H of a finite group G is called ℙ-subnormal in G whenever H either coincides with G or is connected to G by a chain of subgroups of prime indices. If every Sylow subgroup of G is ℙ-subnormal in G then G is called a w-supersoluble group. We obtain some properties of ℙ-subnormal subgroups and the groups that are products of two ℙ-subnormal subgroups, in particular, of ℙ-subnormal w-supersoluble subgroups.