Finite groups with ℱ-subnormal conditions

Let ? be a subgroup-closed saturated formation. A finite group G is called an ?_pc-group provided that each subgroup X of G is ?-subabnormal in the ?-subnormal closure of X in G. Let ?_pc be the class of all ?_pc-groups. We study some properties of ? _pc-groups and describe the structure of ?_pc-groups when ? is the class of all soluble π-closed groups, where π is a given nonempty set of prime numbers.     

Bass-Tits minimization of automata, quotients of trees and diameters

Let X be a tree and let G=Aut(X), Bass and Tits have given an algorithm to construct the ‘ultimate quotient’ of X by G starting with any quotient of X, an ‘edge-indexed’ graph. Using a sequence of integers that we compute at consecutive steps of the Bass-Tits (BT) algorithm, we give a lower bound on the diameter of the ultimate quotient of a tree by its automorphism group. For a tree X with finite quotient, this gives a lower bound on the minimum number of generators of a uniform X-lattice whose quotient graph coincides with G?X. This also gives a criterion to determine if the ultimate quotient of a tree is infinite. We construct an edge-indexed graph (A,i) for a deterministic finite state automaton and show that the BT algorithm for computing the ultimate quotient of (A,i) coincides with state minimizing algorithm for finite state automata. We obtain a lower bound on the minimum number of states of the minimized automaton. This gives a new proof that language for the word problem in a finitely generated group is regular if and only if the group is finite, and a new proof that the language of the membership problem for a subgroup is regular if and only if the subgroup has finite index.     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

Primitive permutation groups with a regular subgroup

This paper starts the classification of the primitive permutation groups (G,Ω) such that G contains a regular subgroup X. We determine all the triples (G,Ω,X) with soc(G) an alternating, or a sporadic or an exceptional group of Lie type. Further, we construct all the examples (G,Ω,X) with G a classical group which are known to us. Our particular interest is in the 8-dimensional orthogonal groups of Witt index 4. We determine all the triples (G,Ω,X) with . In order to obtain all these triples, we also study the almost simple groups G with GPΩ_2n+1(q). The case GU_n(q) is started in this paper and finished in [B. Baumeister, Primitive permutation groups of unitary type with a regular subgroup, Bull. Belg. Math. Soc. 112 (5) (2006) 657–673]. A group X is called a Burnside-group (or short a B-group) if each primitive permutation group which contains a regular subgroup isomorphic to X is necessarily 2-transitive. In the end of the paper we discuss B-groups.     

On the diameter of the intersection graph of a finite simple group

Let G be a finite group. The intersection graph Δ_G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected.     

ON ALGEBRAIC VOLUME DENSITY PROPERTY

A smooth affine algebraic variety X equipped with an algebraic volume form ω has the algebraic volume density property (AVDP) if the Lie algebra generated by complete algebraic vector fields of ω-divergence zero coincides with the space of all algebraic vector fields of ω-divergence zero. We develop an effective criterion of verifying whether a given X has AVDP. As an application of this method we establish AVDP for any homogeneous space X = G/R that admits a G-invariant algebraic volume form where G is a linear algebraic group and R is a closed reductive subgroup of G.     

On The Lattice of Regular Transitive Subgroup Functors

We study the structure of the lattice Reg_trϰ) of all regular transitive subgroup ϰ-functors. We describe all hereditary formations ϰ for which the width of Reg_tr(ϰ) is finite and does not exceed |π(ϰ)|, where fy(ϰ) is the set of all prime divisors of the orders of the groups in ϰ.     

Finite Groups Whose <Emphasis Type="Italic">n</Emphasis>-Maximal Subgroups Are Modular

Let G be a finite group. If M_n< M_n?1< · · · < M₁< M₀ = G with Mi a maximal subgroup of M_i?1 for all i = 1,..., n, then M_n (n > 0) is an n-maximal subgroup of G. A subgroup M of G is called modular provided that (i) 〈X,M ∩ Z〉 = 〈X,M〉 ∩ Z for all X ≤ G and Z ≤ G such that X ≤ Z, and (ii) 〈M,Y ∩ Z〉 = 〈M,Y 〉 ∩ Z for all Y ≤ G and Z ≤ G such that M ≤ Z. In this paper, we study finite groups whose n-maximal subgroups are modular.     

Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups

Let P be a subgroup of a Sylow subgroup of a finite group G. If P is a Sylow subgroup of some normal subgroup of G then P is called normally embedded in G. We establish tests for a finite group G to be p-supersoluble provided that every maximal subgroup of a Sylow p-subgroup of X is normally embedded in G. We study the cases when X is a normal subgroup of G, X = O_p',p(H), and X = F*(H) where H is a normal subgroup of G.     

Strong homology and dimension

We show that strong homology groups H̄_p(X; G) of a space X vanish if p is greater than the shape dimension sd X. For p=sd X, H̄_p(X; G) coincides with the Čech homology groups Ȟ_p(X; G). We also show that there exist 1-dimensional spaces, which do not admit 1-dimensional ANR-resolutions. Therefore, the vanishing of H̄_p(X; G) for p>dim X is a nontrivial fact.     

Classifying Cubic Semisymmetric Graphs of Order 18 p n

A graph X is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut (X) acting transitively on its edge set but not on its vertex set. In the case of G = A, we call X a semisymmetric graph. Let p be a prime. It was shown by Folkman (J Comb Theory 3:215–232, 1967) that a regular edge-transitive graph of order 2p or 2p ² is necessarily vertex-transitive. The smallest semisymmetric graph is the Folkman graph. In this study, we classify all connected cubic semisymmetric graphs of order 18p ⁿ , where p is a prime and \({n \geq 1}\) .     

On permuteral subgroups in finite groups

The permutizer of a subgroup H in a group G is defined as the subgroup generated by all cyclic subgroups of G that permute with H. Call H permuteral in G if the permutizer of H in G coincides with G; H is called strongly permuteral in G if the permutizer of H in U coincides with U for every subgroup U of G containing H. We study the finite groups with given systems of permuteral and strongly permuteral subgroups and find some new characterizations of w-supersoluble and supersoluble groups.     

Algebraic density property of homogeneous spaces

Let X be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that X is equipped with several fixed point free nondegenerate SL₂-actions satisfying some mild additional assumption. Then we prove that the Lie algebra generated by completely integrable algebraic vector fields on X coincides with the space of all algebraic vector fields. In particular, we show that apart from a few exceptions this fact is true for any homogeneous space of form G/R where G is a linear algebraic group and R is a closed proper reductive subgroup of G.     

Every 2-group with all subgroups normal-by-finite is locally finite

A group G has all of its subgroups normal-by-finite if H/H _G is finite for all subgroups H of G. The Tarski-groups provide examples of p-groups (p a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a 2-group with every subgroup normal-by-finite is locally finite. We also prove that if |H/H _G | 6 2 for every subgroup H of G, then G contains an Abelian subgroup of index at most 8.     

Finite groups with some given systems of Xm‐semipermutable subgroups

Let X be a nonempty subset of a group G. We call a subgroup A of G an X_m‐semipermutable subgroup of G if A has a minimal supplement T in G such that for every maximal subgroup M of any Hall subgroup T₁ of T there exists an element x ∈ X such that AM^x = M^xA. In this paper, we study the structure of finite groups with some given systems of X_m‐semipermutable subgroups (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Frobenius splitting and geometry of G-Schubert varieties

Let X be an equivariant embedding of a connected reductive group G over an algebraically closed field k of positive characteristic. Let B denote a Borel subgroup of G. A G-Schubert variety in X is a subvariety of the form diag(G)⋅V, where V is a B×B-orbit closure in X. In the case where X is the wonderful compactification of a group of adjoint type, the G-Schubert varieties are the closures of Lusztig's G-stable pieces. We prove that X admits a Frobenius splitting which is compatible with all G-Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any G-Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that G-Schubert varieties have nice singularities we present an example of a nonnormal G-Schubert variety in the wonderful compactification of a group of type G₂. Finally we also extend the Frobenius splitting results to the more general class of R-Schubert varieties.     

The Hopfian property of n-periodic products of groups

LetH be a subgroup of a groupG. A normal subgroupN _H ofH is said to be inheritably normal if there is a normal subgroup N _G of G such that N _H = N _G ∩ H. It is proved in the paper that a subgroup $N_{G_i }$ of a factor G _i of the n-periodic product Π _i∈I ⁿ G _i with nontrivial factors G _i is an inheritably normal subgroup if and only if $N_{G_i }$ contains the subgroup G _i ⁿ . It is also proved that for odd n ≥ 665 every nontrivial normal subgroup in a given n-periodic product G = Π _i∈I ⁿ G _i contains the subgroup G ⁿ . It follows that almost all n-periodic products G = G _{1 *} ⁿ G ₂ are Hopfian, i.e., they are not isomorphic to any of their proper quotient groups. This allows one to construct nonsimple and not residually finite Hopfian groups of bounded exponents.     

Free groups and ends of graphs

Let X be a connected graph. An automorphism of X is said to be parabolic if it leaves no finite subset of vertices in X invariant and fixes precisely one end of X and hyperbolic if it leaves no finite subset of vertices in X invariant and fixes precisely two ends of X. Various questions concerning dynamics of parabolic and hyperbolic automorphisms are discussed.The set of ends which are fixed by some hyperbolic element of a group G acting on X is denoted by ?(G). If G contains a hyperbolic automorphism of X and G fixes no end of X, then G contains a free subgroup F such that ?(F) is dense in ?(G) with respect to the natural topology on the ends of X.As an application we obtain the following: A group which acts transitively on a connected graph and fixes no end has a free subgroup whose directions are dense in the end boundary.     

On G-arc-regular dihedrants and regular dihedral maps

A graph Γ is said to be G-arc-regular if a subgroup $G \le\operatorname{\mathsf{Aut}}(\varGamma)$ acts regularly on the arcs of Γ. In this paper connected G-arc-regular graphs are classified in the case when G contains a regular dihedral subgroup D _2n of order 2n whose cyclic subgroup C _n ≤D _2n of index 2 is core-free in G. As an application, all regular Cayley maps over dihedral groups D _2n , n odd, are classified.