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.

On minimal nonPN-groups

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 Note on the Solvablity of Groups

Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K M. We call the group M ∩ K/L a c-section of M. And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M. For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L 2 (p) or Z q , where p and q are primes, and p ≡ ±1(mod 8). This result answer a question posed by ref. [12].

A note on the solvability of groups

Journal of Algebra, 2006

Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K M. We call the group M ∩ K/L a c-section of M. And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M. For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L 2 (p) or Z q , where p and q are primes, and p ≡ ±1(mod 8). This result answer a question posed by ref. [12].

IRJET- On Some Minimal S-Quasinormal Subgroups of Finite Groups

IRJET, 2020

A subgroup H of a group G is permutable subgroup of G if for all subgroups S of G the following condition holds SH = HS < S,H >. A subgroup H is S-quasinormal in G if it permutes with every Sylow subgroup of G. In this article we study the influence of S-quasinormality of subgroups of some subgroups of G on the super-solvability of G.

Non-subnormal subgroups of groups

Journal of Pure and Applied Algebra, 2013

According to Möhres's Theorem an arbitrary group whose proper subgroups are all subnormal (or a group without non-subnormal proper subgroups) is solvable. In this paper we generalize Möhres's Theorem, by proving that every group with at most 56 nonsubnormal subgroups is solvable. Also we show that the derived length of a solvable group with a finite number k of non-n-subnormal subgroups is bounded in terms of n and k.