On p-nilpotency of finite groups with some subgroups π-quasinormally embedded

Summary A subgroup H of a group G is said to be π-quasinormal in G if it permutes with every Sylow subgroup of G, and H is said to be π-quasinormally embedded in G if for each prime dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some π-quasinormal subgroups of G. We characterize p-nilpotentcy of finite groups with the assumption that some maximal subgroups, 2-maximal subgroups, minimal subgroups and 2-minimal subgroups are π-quasinormally embedded, respectively.     

Parabolic and quasiparabolic subgroups of free partially commutative groups

Let Γ be a finite graph and G be the corresponding free partially commutative group. In this paper we study subgroups generated by vertices of the graph Γ, which we call canonical parabolic subgroups. A natural extension of the definition leads to canonical quasiparabolic subgroups. It is shown that the centralisers of subsets of G are the conjugates of canonical quasiparabolic centralisers satisfying certain graph theoretic conditions.     

Pairwise -connected Products of Certain Classes of Finite Groups

Abstract

Subgroups A and B of a finite group are said to be 𝒩-connected if the subgroup generated by elements x and y is a nilpotent group, for every pair of elements x in A and y in B. The behaviour of finite pairwise permutable and 𝒩-connected products are studied with respect to certain classes of groups including those groups where all the subnormal subgroups permute with all the maximal subgroups, the so-called SM-groups, and also the class of soluble groups where all the subnormal subgroups permute with all the Carter subgroups, the so-called C-groups.     

The geometry of k-transvection groups

Let k be a (commutative) field and G a group, then a conjugacy class of Abelian subgroups of G is called a class of k-transvection subgroups in G if and only if it generates G and any two elements of the class either commute or are full unipotent subgroups of the group they generate and which is isomorphic to (P)SL₂(k).In this paper we study the geometry of k-transvection groups. Given a class of k-transvection groups Σ, we consider a partial linear space whose points are the elements of Σ, and whose lines correspond to the groups generated by two noncommuting elements from Σ. We derive several properties of this partial linear space. These properties are used to give a characterization of the geometries of k-transvection groups and provide a classification of groups generated by k-transvection subgroups.     

Maximal 2-Local Geometries for the Symmetric Groups

Abstract

The set 𝒩_max (G, T) consisting of all maximal 2-local subgroups of G = Sym(n) which contain T, a Sylow 2-subgroup of G, is investigated. In addition to determining the structure of the subgroups in 𝒩_max (G, T), the simplicial sets of maximal rank are classified.     

Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces

The aim of this paper is to count subgroups of a given index in the fundamental group of an orientable S ¹-bundle over a compact surface. The number of subgroups of index n turns out to be independent of the orientability of the base surface 𝔣, closed or bordered, and is expressed as a linear combination of the numbers of surface subgroups of indices m = n/l l∣n. For a closed base surface of characteristic χ the respective coefficients are equal to l ^–χm+2 or vanish depending on l,n and the Euler number of the S ¹ -bundle.     

Conjugacy of net subgroups of linear groups

For the full linear group over a matrix-local ring whose quotient by the Jacobson radical is not the field of two elements, we settle the question of the conjugacy ofD -net subgroups (Ref. Zh. Mat., 1977, 2A280). TwoD -net subgroups are conjugate if and only if theD -nets defining them are similar (i.e., can be transformed into each other by a permutation matrix). An analogous result is obtained forD -net subgroups of the symplectic group over a commutative local ring whose residue field contains more than three elements.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 86, pp. 11–18, 1979.     

Sylow Products and the Solvable Residual

We study the connection between products of Sylow subgroups of a finite group G and the solvable residual of G. Let Π(𝒫) be a product of Sylow subgroups of G such that each prime divisor of |G| is represented exactly once in Π(𝒫). We prove that there exists a unique normal subgroup N of G which is minimal subject to the requirement Π(𝒫) N = G. Furthermore, N is perfect, and the product of all of these subgroups is the solvable residual of G. We also prove that the solvable residual of G is generated by all elements which arise from non-trivial factorizations of 1 _G in such products of Sylow subgroups.     

Nilpotent Injectors in General Linear Groups

In a recent paper, A. Bialostocki (Israel J. Math.41 (1982), 261-273) has defined a nilpotent injector in an arbitrary finite group G to be a maximal nilpotent subgroup of G, containing a subgroup H of G of maximal order satisfying class (H) ≤2. In the present paper, the author determines the nilpotent injectors of GL(n, q) and shows that they form a unique conjugacy class of subgroups of GL(n, q). It is also proved that if n ≠ 2 or n = 2 and q ≠ 9 is not a Fermat prime >3, then the nilpotent injectors of GL(n, q) are the nilpotent subgroups of maximal order.     

Twisted Equivariant KK-Theory and the Baum–Connes Conjecture for Group Extensions

In this paper we study the stability of the Baum–Connes conjecture with coefficients undertaking group extensions. For this, it is necessary to extend Kasparov's equivariant KK-theory to an equivariant theory for twisted actions of groups on C ^*-algebras. As a consequence of our stability results, we are able to reduce the problem of whether closed subgroups of connected groups satisfy the Baum–Connes conjecture, with coefficients to the special case of center-free semi-simple Lie groups.     

Groups with many subgroups having polycyclic-by-finite layers or derived subgroup

A group is called (PF)L if the subgroups generated by its elements having same order (finite or infinite) are polycyclic-by-finite. In the present paper we prove that a group is locally graded minimal non-((PF)L∪(𝔓𝔉)𝔄) if, and only if, it is non-perfect minimal non-FC, where (𝔓𝔉)𝔄 denotes the class of (polycyclic-by-finite)-by-abelian groups. We prove also that a group of infinite rank whose proper subgroups of infinite rank are in ((PF)L∪(𝔓𝔉)𝔄) is itself in ((PF)L∪(𝔓𝔉)𝔄) provided that it is locally (soluble-by-finite) without simple homomorphic images of infinite rank. Our last result concerns groups that satisfy the minimal condition on non-((PF)L∪(𝔓𝔉)𝔄)-subgroups.     

Some subgroups of the full linear group

Further observations are made on the author's earlier paper (Ref. Zh. Mat., 1977, 5A284) dealing with the lattice H of all subgroups of the full linear group GL(n, K) over a field K that contain the group K of diagonal matrices. It is noted, for example, that for an infinite field K all subgroups inD(n, K) are algebraic; a subgroup in H is connected if and only if it is a net subgroup; the lattice of all connected subgroups in H is isomorphic to the lattice of all marked topologies onn points; any subgroup H in H is a semidirect product H=A·H_o of a maximal connected normal subgroup H_o of H and a finite group A of, permutation matrices.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 42–46, 1977.     

Embedding properties of nonprimary subgroups of the symmetric group of degree eight

The paper is devoted to studying some subgroup embedding properties for 110 subgroups of the symmetric group of degree 8. We deal with those nonprimary subgroups whose degree is exactly eight. The results were obtained by using of the computer algebra system GAP. They are summarized in a table. It was found that every polynormal subgroup is also paranormal. Bibliography: 12titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 124–128.     

Groups with less than <Emphasis Type="Italic">n</Emphasis> subgroups of index <Emphasis Type="Italic">n</Emphasis>

We characterise (residually-finite) groups which possess less than n subgroups of index n for almost all n ∈ ℕ.     

Coradicals of subnormal subgroups

IfF is a nonempty formation, then theF-coradical of a finite group G is the intersection of all those normal subgroups N of G for which G / N F. We study the structure of theF-coradical of a group generated by two subnormal subgroups of a finite group. The results are used to reveal properties sufficient for theF-coradicals of subnormal subgroups to be permutable, forF a composition formation.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 493–513, September-October, 1995.     

Isomorphisms between lattices of almost normal subgroups

A subgroupH of a groupG is said to bealmost normal inG if it has only finitely many conjugates inG. The setan(G) of almost normal subgroups ofG is a sublattice of the lattice of all subgroups ofG. Isomorphisms between lattices of almost normal subgroups ofFC-soluble groups are considered in this paper. In particular, properties of images of normal subgroups under such an isomorphism are investigated.     

On minimal non-MSN-groups

A finite group G is called an MSN-group if all maximal subgroups of the Sylow subgroups of G are subnormal in G. In this paper, we determinate the structure of non-MSN-groups in which all of whose proper subgroups are MSN-groups.