On supersolvable groups whose maximal subgroups of the Sylow subgroups are subnormal (original) (raw)
Groups with maximal subgroups of Sylow subgroups normal
Israel Journal of Mathematics, 1982
This paper characterizes those finite groups with the property that maximal subgroups of Sylow subgroups are normal. They are all certain extensions of nilpotent groups by cyclic groups.
On two classes of finite supersoluble groups
Communications in Algebra, 2017
Let Z be a complete set of Sylow subgroups of a nite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called Z-S-semipermutable if H permutes with every Sylow p-subgroup of G in Z for all p / ∈ π(H); H is said to be Z-S-seminormal if it is normalized by every Sylow p-subgroup of G in Z for all p / ∈ π(H). The main aim of this paper is to characterize the Z-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in Z are Z-S-semipermutable in G and the Z-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in Z are Z-S-seminormal in G.
Journal of Algebra, 1980
A well-known theorem of Wielandt states that a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G. The structure of a nonnilpotent group, each of whose proper subgroups is nilpotent, has been analyzed by Schmidt and R6dei [5, Satz 5.1 and Satz 5.2, pp. 280-281]. In [1], Buckley investigated the structure of a PN-group (i.e., a finite group in which every minimal subgroup is normal), and proved (i) that a PN-group of odd order is supersolvable, and (ii) that certain factor groups of a PN-group of odd prime power order are also PN-groups. Earlier, Gaschiitz and It5 [5, Satz 5.7, p. 436] had proved that the commutator subgroup of a finite PN-group is p-nilpotent for each odd prime p. This paper is a sequel to [9] and our object here is to prove the following statement. THEOREM. If G is a finite nonPN-group, each of whose proper subgroups is a PN-group, then one of the following statements is true: (a) G is the dihedral group of order 8.
The influence of SS-quasinormality of some subgroups on the structure of finite groups
Journal of Algebra, 2008
The following concept is introduced: a subgroup H of the group G is said to be SS-quasinormal (Supplement-Sylow-quasinormal) in G if H possesses a supplement B such that H permutes with every Sylow subgroup of B. Groups with certain SS-quasinormal subgroups of prime power order are studied. For example, fix a prime divisor p of |G| and a Sylow p-subgroup P of G, let d be the smallest generator number of P and M d (P ) denote a family of maximal subgroups P 1 , . . . , P d of P satisfying d i=1 (P i ) = Φ(P ), the Frattini subgroup of P . Assume that the group G is p-solvable and every member of some fixed M d (P ) is SS-quasinormal in G, then G is p-supersolvable.
A class of generalized supersoluble groups
Publicacions Matemàtiques, 2005
This paper is devoted to the study of groups G in the universe cL of all radical locally finite groups with min-p for all primes p such that every δ-chief factor of G is either a cyclic group of prime order or a quasicyclic group. We show that within the universe cL this class of groups behaves very much as the class of finite supersoluble groups.
Minimality and locally defined classes of groups
Ricerche di Matematica, 2009
The class of groups G for which every subgroup of P ∈ Syl p (G) is normal (permutable) in N G (P) is called C p (X p). A finite solvable group possesses a transitive normality (permutability) relation if and only if it is a C p-group (X p-group) for all primes p. The classes T , PT , and PST denote, respectively, the classes of groups in which normality, permutability, and S-permutability are transitive relations. Our main result shows that the minimal non-C p-groups and the minimal non-X p-groups, respectively, are just the minimal non-T-groups and the minimal non-PT-groups. In addition, we arrive a new characterization of the solvable PT-groups and the solvable PST-groups.
Finite Groups with SS-Quasinormal Subgroups
International Journal of Algebra, 2010
A subgroup H of a group G is said to be SS-quasinormal (supplement-Sylow-quasinormal) in G if there is a supplement B of H to G such that H is permutable with every Sylow subgroup of B. In this paper we investigate the influence of SS-quasinormality of minimal subgroups or 2-minimal subgroups of finite group and extent the result of A. Carocca and some well-known results.
Maximal subgroups and PST-groups
Central European Journal of Mathematics, 2013
A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions of Kaplan’s results, which enables a better understanding of the relationships between these classes.