New characterizations of p-nilpotency of finite groups

Suppose that [Formula: see text] is a finite group and [Formula: see text] is a subgroup of [Formula: see text]. [Formula: see text] is said to be an [Formula: see text]-quasinormal subgroup of [Formula: see text] if there is a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] permutes with every Sylow subgroup of [Formula: see text]. In this note, we fix in every non-cyclic Sylow subgroup [Formula: see text] of [Formula: see text] some subgroup [Formula: see text] satisfying [Formula: see text] and study the [Formula: see text]-nilpotency of [Formula: see text] under the assumption that every subgroup [Formula: see text] of [Formula: see text] with [Formula: see text] is [Formula: see text]-quasinormal in [Formula: see text]. The Frobenius theorem is generalized.

Theoretical Researches about u-Maximal Subgroups and Its Applications in Charactering IntuG

Let G be a finite group and u be the class of all finite supersoluble groups. A supersoluble subgroup U of G is called u-maximal in G if for any supersoluble subgroup V of G containing U, V=U. Moreover, IntuG is the intersection of all u-maximal subgroups of G. This paper obtains some new criteria on IntuG, by assuming that some subgroups of G are either Φ-I-supplemented or Φ-I-embedded in G. Here, a subgroup H of G is called Φ-I-supplemented in G if there exists a subnormal subgroup T of G such that G=HT and H∩THG/HG≤ΦH/HGIntuG and Φ-I-embedded in G if there exists a S-quasinormal subgroup T of G such that HT is S-quasinormal in G and H∩THG/HG≤ΦH/HGIntuG.

On s-semipermutable or s-quasinormally Embedded Subgroups of Finite Groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be s-semipermutable in G if HGp = GpH for any Sylow p-subgroup Gp of G with (p, |H|) = 1; H is said to be s-quasinormally embedded in G if for each prime p dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal subgroup of G. In every non-cyclic Sylow subgroup P of G we fix some subgroup D satisfying 1 < |D| < |P| and study the structure of G under the assumption that every subgroup H of P with |H| = |D| is either s-semipermutable or s-quasinormally embedded in G. Some recent results are generalized and unified.

On p-supersolubility and solubility of finite groups

Let 𝔉 be a class of groups. A subgroup H of G is said to be weakly 𝔉s-quasinormal in G if G has an S-quasinormal subgroup T such that HT is S-quasinormal in G and (H ∩ T) HG/HG ≤ Z𝔉(G/HG), where Z𝔉(G/HG) is the 𝔉-hypercenter of G/HG. In this paper, we investigate further the influence of weakly 𝔉s-quasinormality of some subgroups on the structure of finite groups. Some new characterizations about p-supersolubility and solubility of finite groups are obtained.

New characterizations of p-soluble and p-supersoluble finite groups

Let G be a finite group and H a subgroup of G. H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G and HsG the intersection of all S-quasinormal subgroups of G containing H. The symbol |G|p denotes the order of a Sylow p-subgroup of G. We prove the followingTheorem A. Let G be a finite group and p a prime dividing |G|. Then G is p-supersoluble if and only if for every cyclic subgroup H ofḠ (G) of prime order or order 4 (if p = 2), Ḡhas a normal subgroup T such thatHsḠandH∩T=HsḠ∩T.Theorem B. A soluble finite group G is p-supersoluble if and only if for every 2-maximal subgroup E of G such that Op′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T with cyclic Sylow p-subgroups such that EsG = ET and |E ∩ T|p = |EsG ∩ T|p.Theorem C. A finite group G is p-soluble if for every 2-maximal subgroup E of G such that Op′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T such that EsG = ET and |E ∩ Tp = |EsG ∩ T|p.

FINITE GROUPS WITH WEAKLY S-QUASINORMALLY EMBEDDED SUBGROUPS

Let H be a subgroup of a group G. A subgroup H of G is said to be S-quasinormally embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. We say that H is weakly S-quasinormally embedded in G if there exists a normal subgroup T of G such that HT ⊴ G and H ∩ T is S-quasinormally embedded in G. In this paper, we investigate further the influence of weakly S-quasinormally embedded subgroups on the structure of finite groups. A series of known results are generalized.

GENERALISING QUASINORMAL SUBGROUPS

In Cossey and Stonehewer [‘On the rarity of quasinormal subgroups’, Rend. Semin. Mat. Univ. Padova125 (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 pn containing a quasinormal subgroup H of exponent pn−1 such that the nontrivial quasinormal subgroups of G lying in H can have exponent only p, pn−1 or, when n≥4 , pn−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.

FINITE GROUPS WITH SOME SEMI-p-COVER-AVOIDING OR S-QUASINORMALLY EMBEDDED SUBGROUPS

A subgroup H of a group G is called S-quasinormally embedded in G if, for each prime p dividing the order of H, a Sylow p-subgroup of H is a Sylow p-subgroup of an S-quasinormal subgroup of G. H is said to be semi-p-cover-avoiding in G if there is a chief series 1 = G0 < G1 < ⋯ < Gt = G of G such that, for every i = 1, 2, ⋯, t, if Gi/Gi-1 is a p-chief factor, then H either covers or avoids Gi/Gi-1. We give the structure of a finite group G in which some subgroups of G with prime-power order are either semi-p-cover-avoiding or S-quasinormally embedded in G.

FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

On quasinormal subgroups of certain finitely generated groups

A subgroup Q of a group G is called quasinormal in G if Q permutes with every subgroup of G. Of course a quasinormal subgroup Q of a group G may be very far from normal. In fact, examples of Iwasawa show (for a convenient reference see [8]) that we may have Q core-free and the normal closure QG of Q in G equal to G so that Q is not even subnormal in G. We note also that the core of Q in G, QG, is of infinite index in QG in this example. If G is finitely generated then any quasinormal subgroup Q is subnormal in G [8] and although Q is not necessarily normal in G we have that |QG:Q| is finite and |QG:Q| is a nilpotent group of finite exponent [5].