Finite Groups in which τ\tauτ-Quasinormality is a Transitive Relation (original) (raw)

On weakly τ-quasinormal subgroups of finite groups

Acta Mathematica Hungarica, 2015

Let G be a nite group and H a subgroup of G. We say that: (1) H is τ-quasinormal in G if H permutes with all Sylow subgroups Q of G such that |Q|, |H| = 1 and |H|, |Q G | = 1; (2) H is weakly τ-quasinormal in G if G has a subnormal subgroup T such that HT = G and T ∩ H H τ G , where H τ G is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. Our main result here is the following. Let F be a saturated formation containing all supersoluble groups and let X E be normal subgroups of a group G such that G/E ∈ F. Suppose that every non-cyclic Sylow subgroup P of X has a subgroup D such that 1 < |D| < |P | and every subgroup H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is non-Abelian) not having a supersoluble supplement in G is weakly τ-quasinormal in G. If X is either E or F * (E), then G ∈ F.

Characterization of Finite Groups With Some S-quasinormal Subgroups

Monatshefte für Mathematik, 2005

A subgroup of a finite group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we give a characterization of a finite group G under the assumption that every subgroup of the generalized Fitting subgroup of prime order is S-quasinormal in G.

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.

On the Rarity of Quasinormal Subgroups

Rendiconti del Seminario Matematico della Università di Padova, 2011

For each prime p and positive integer n, Berger and Gross have defined a finite p-group G HX, where H is a core-free quasinormal subgroup of exponent p nÀ1 and X is a cyclic subgroup of order p n. These groups are universal in the sense that any other finite p-group, with a similar factorisation into subgroups with the same properties, embeds in G. In our search for quasinormal subgroups of finite p-groups, we have discovered that these groups G have remarkably few of them. Indeed when p is odd, those lying in H can have exponent only p, p nÀ2 or p nÀ1. Those of exponent p are nested and they all lie in each of those of exponent p nÀ2 and p nÀ1 .

Quasinormal subgroups of finite p-groups

Note Di Matematica, 2011

The distribution of quasinormal subgroups within a group is not particularly well understood. Maximal ones are clearly normal, but little is known about minimal ones or about maximal chains. The study of these subgroups in finite groups quickly reduces to p-groups. Also within an abelian quasinormal subgroup, others (quasinormal in the whole group) abound. But in non-abelian quasinormal subgroups, the existence of others can be dramatically rare.

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.

S-quasinormallity of finite groups

Frontiers of Mathematics in China, 2010

Let d be the smallest generator number of a finite p-group P, and let M d (P ) = {P 1 , . . . , P d } be a set of maximal subgroups of P such that ∩ d i=1 P i = Φ(P ). In this paper, the structure of a finite group G under some assumptions on the S-quasinormally embedded or SS-quasinormal subgroups in M d (P ), for each prime p, and Sylow p-subgroups P of G is studied.

On SS-quasinormal and S-quasinormally embedded subgroups of finite groups

Mathematical Notes, 2014

A subgroup H of a group G is said to be an SS-quasinormal (Supplement-Sylow-quasinormal) subgroup if there is a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B. A subgroup H of a group G is said to be S-quasinormally embedded in G if for every Sylow subgroup P of H, there is an S-quasinormal subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain SS-quasinormal or S-quasinormally embedded subgroups of prime power order are studied.