X-permutable maximal subgroups of Sylow subgroups of finite groups

We study finite groups whose maximal subgroups of Sylow subgroups are permutable with maximal subgroups. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1299–1309, October, 2006.     

On second maximal subgroups of Sylow subgroups of finite groups

Finite groups in which the second maximal subgroups of the Sylow p-subgroups, p a fixed prime, cover or avoid the chief factors of some of its chief series are completely classified.     

Groups with dense normal subgroups

A group is said to have dense normal subgroups, if each non-empty open interval in its lattice of subgroups contains a normal subgroup. The structure of this and related classes of groups is investigated. Typical results are: an infinite group with dense ascendant subgroups is locally nilpotent: a nontorsion group with dense normal subgroups is abelian, etc.     

A family of maximal subgroups containing the Sylow subgroups and some solvability conditions

Archiv der Mathematik -     

A note on s-conditional permutability of maximal subgroups of Sylow subgroups of finite groups

Monatshefte für Mathematik - In this note, we point out an error in the paper “On s-conditional permutability of maximal subgroups of Sylow subgroups of finite groups” (Monatsh...     

Groups with two Sylow numbers are the product of two nilpotent Hall subgroups

We have recently found a characterization of the existence of nilpotent Hall subgroups in a finite group. Using this result, we prove the statement in the title. This is a strong form of a conjecture of J. Zhang.     

Groups with many normal or self-normalizing subgroups

In this paper groups in which the set Σ of the normal or self-normalizing subgroups is large will be studied. In particular it will be characterized locally graded groups satisfying the minimal condition on subgroups which do not belong to Σ and locally finite groups for which the set Σ is dense in the lattice of all subgroups.     

Self-normalizing Sylow subgroups

Using the classification of finite simple groups we prove the following statement: Let 3$"> be a prime, a group of automorphisms of -power order of a finite group , and a -invariant Sylow -subgroup of . If is trivial, then is solvable. An equivalent formulation is that if has a self-normalizing Sylow -subgroup with 3$"> a prime, then is solvable. We also investigate the possibilities when .

     

Groups with many normal or self-normalizing subgroups

In this paper groups in which the set Σ of the normal or self-normalizing subgroups is large will be studied. In particular it will be characterized locally graded groups satisfying the minimal condition on subgroups which do not belong to Σ and locally finite groups for which the set Σ is dense in the lattice of all subgroups.     

Finite groups with seminormal Sylow subgroups

In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.     

Infinite groups with Sylow permutable subgroups

If a subgroup H of a periodic group G satisfies HP = PH for all Sylow subgroups P of G, then we call H a Sylow-permutable, or S-permutable, subgroup of G. It is well known that S-permutability is not a transitive relation. In this paper, we study infinite periodic groups in which the relation to be S -permutable is transitive (PST-groups) and infinite periodic groups whose ascendant subgroups are S-permutable (ASP-groups).