On SS-quasinormal and S-quasinormally embedded subgroups of finite groups (original) (raw)

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.

Finite groups with normally embedded subgroups

Journal of group theory, 2010

A subgroup H of the finite group G is said to be quasinormally (resp. Squasinormally) embedded in G if for every Sylow subgroup P of H, there is a quasinormal (resp. S-quasinormal) subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain quasinormally (resp. S-quasinormally) embedded subgroups of prime-power order are studied. For example, if a group G has a normal subgroup H such that G=H A F and such that for each Sylow subgroup P of H, every member in some M d ðPÞ is quasinormally embedded in G, then G A F: here M d ðPÞ is a set of maximal subgroups of P with intersection the Frattini subgroup.

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 .

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.

On s-quasinormal and c-normal subgroups of a finite group

Acta Mathematica Sinica, English Series, 2008

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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.

Generalising Quasinormal Subgroups

Bulletin of the Australian Mathematical Society, 2012

In Cossey and Stonehewer ['On the rarity of quasinormal subgroups', Rend. Semin. Mat. Univ. Padova 125 (2011), 81-105] it is shown that for any odd prime p and integer n ≥ 3, there is a finite p-group G of exponent p n containing a quasinormal subgroup H of exponent p n−1 such that the nontrivial quasinormal subgroups of G lying in H can have exponent only p, p n−1 or, when n ≥ 4, p n−2. Thus large sections of these groups are devoid of quasinormal subgroups. The authors ask in that paper if there is a nontrivial subgroup-theoretic property X of finite p-groups such that (i) X is invariant under subgroup lattice isomorphisms and (ii) every chain of X-subgroups of a finite p-group can be refined to a composition series of X-subgroups. Failing this, can such a chain always be refined to a series of X-subgroups in which the intervals between adjacent terms are restricted in some significant way? The present work embarks upon this quest. 2010 Mathematics subject classification: primary 20E07; secondary 20E15.

The Influence of S-Embedded Subgroups on the Structure of Finite Groups

2015

Let H be a subgroup of a group G. H is said to be S-embedded in G if G has a normal subgroup T such that HT is an S-permutable subgroup of G and H \ T HsG, where HsG denotes the subgroup generated by all those subgroups of H which are S-permutable in G. In this paper, we investigate the inuence of minimal S-embedded subgroups on the structure of nite groups. We determine the structure of nite groups with some minimal S- embedded subgroups. We also give some new characterizations of p-nilpotency of nite groups in terms of the S-embedding property. As applications, some previously known results are generalized. Keywords: Finite groups, S-embedded subgroups, the generalized Fitting subgroups, soluble groups, p-nilpotent groups. MSC(2010): Primary: 20D10; Secondary: 20D15, 20D20, 20D25.

The influence of SS-quasinormality of some subgroups

Proof. All the non-abelian subgroups of G are G and Q 8 . Furthermore, G = Q 8 and Q 8 = Z (Q 8 ), the center of Q 8 , which are normal in G. Hence D(G) = G, and G is non-abelian. 2 Example 1.3. As Aut(Q 8 ) ∼ = S 4 and D 8 S 4 , we have the semidirect product G = [Q 8 ]D 8 . Then G is a 2-group of order 2 6 and D(G) < G.