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     

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 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.     

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.     

Modularity in the lattice of Σ-permutable subgroups

For a Hall system of a finite solvable group G, it is known that the set of -permutable subgroups is a sublattice of the subgroup lattice of G. We investigate the class SPM of groups in which the lattice is modular. We prove that if is modular, then U V for all (an evidently stronger condition). Both of these phenomena—the modularity of and whether two -permutable subgroups U and V permute with each other—are shown to be determined locally, by what happens at each prime. The class SPM is shown to be quotient closed, but not direct product or subgroup closed.This revised electronic version of the Abstract includes the formulas that were missing in the previous electronic version published online in September 2004.     

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 poset of conjugacy classes of subgroups of π-power index

We investigate tile poset S^π(G)/G of conjugacy clases of subgroups of π-power index in a finite group G. In particular, we are concerned with combinatorial and topological properties of the order complex of S^π(G)/G. We show that the order complex of S^π(G)/G iS homotopic to a join of orbit spaces of order complexes of posets, which bear structural information on the cheif factors of the group. Moreover, for π-solvable groups and in case π = {p} we reveal a shellable subposer of S^π(G)/G of the same homotopy type. This complements the study of the poset S^π(G) of subgroups of π-power index performed in [20]. For the analysis of the order complexes we develop some new lemmata on the topology of order complexes of posets and in the theory of shellability.     

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.     

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.     

Notes on discrete subgroups of Möbius transformations

Jørgensen’s inequality gives a necessary condition for a nonelementary two generator subgroup of $SL(2, {\mathbb C})$ to be discrete. By embedding $SL(2,{\mathbb C})$ into $\hat U(1,1; {\mathbb H})$ , we obtain a new type of Jørgensen’s inequality, which is in terms of the coefficients of involved isometries. We provide an example to show that this result gives an improvement over the classical Jørgensen’s inequality.     

Chevalley–Weil theorem and subgroups of class groups

We prove, under some mild hypothesis, that an ´etale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. This constitutes an “absolute” version of the Chevalley–Weil theorem. Using this result, we are able to generalise the techniques of Mestre, Levin and the second author for constructing and counting number fields with large class group.     

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.     

A characterization of finite simple groups by the orders of solvable subgroups

Let G be a finite group and S be a finite simple group. In this paper, we prove that if G and S have the same sets of all orders of solvable subgroups, then G is isomorphic to S, or G and S are isomorphic to Bn(q), Cn(q), where n≥3 and q is odd. This gives a positive answer to the problem put forward by Abe and Iiyori.     

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...     

Free products of cyclic subgroups in the groupPSL 2(ℂ)

In the paper, families of faithful representations of the free product ofn (n ≥ 2) cyclic groups into the groupPSL ₂(ℂ) are indicated. They are used to generalize a result of Ree and Mendelsohn. We also obtain a series of sufficient conditions under which a 2-generated subgroup 〈A, B〉 of the groupPSL ₂(ℂ) is isomorphic to the free product of its cyclic subgroups 〈A〉 and 〈B〉. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 922–930, June, 2000.