On <Emphasis Type="Italic">s</Emphasis>-semipermutable subgroups of finite groups and <Emphasis Type="Italic">p</Emphasis>-nilpotency

A subgroup H of a group is said to be s-semipermutable in G if it is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. Using the concept of s-semipermutable subgroups, some new characterizations of p-nilpotent groups are obtained and several results are generalized.     

Weakly s-semipermutable subgroups of finite groups

In this paper, we introduce the concept of weakly s-semipermutable subgroups. Let G be a finite group. Using the condition that the minimal subgroups or subgroups of order p ² of a given Sylow p-subgroup of G are weakly s-semipermutable in G, we give a criterion for p-nilpotency of G and get some results about formation.     

On weakly s-semipermutable or ss-quasinormal subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly s-semipermutable in G if there are a subnormal subgroup T of G and an s-semipermutable subgroup \(H_{ssG}\) of G contained in H such that \(G=HT\) and \(H\cap T\le H_{ssG}\); H is said to be an ss-quasinormal subgroup of G if there is a subgroup B of G such that \(G=HB\) and H permutes with every Sylow subgroup of B. We fix in every non-cyclic Sylow subgroup P of G 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 weakly s-semipermutable or ss-quasinormal in G. Some recent results are generalized and unified.

     

The influence of <Emphasis Type="Italic">s</Emphasis>-semipermutable subgroups on the structure of finite groups

A subgroup H of a group G is called s-semipermutable in G if H is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. In this paper, we use s-semipermutable subgroups to determine the structure of finite groups. Some of the previous results are generalized.     

The influence of <Emphasis Type="Italic">X</Emphasis>-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained. This work was supported by National Natural Science Foundation of China (Grant Nos. 10771172, 10771180)     

On s-Permutable Subgroups and p-Nilpotency of Finite Groups

A subgroup H of a finite group G is said to be s-permutable in G if H permutes with every Sylow subgroup of G. In this article, some sufficient conditions for a finite group G to be p-nilpotent are given whenever all subgroups with order p ^m of a Sylow p-subgroup of G are s-permutable for a given positive integer m.     

Some new characterizations of solvable PST-groups

A subgroup H of a finite group G is said to be ??-semipermutable in G if it permutes with all the Sylow subgroups Q of G such that (|H|, |Q|) = 1 and (|H|, |Q ^G |) ?? 1. A rather remarkable result of Lukyanenko and Skiba (Rend Semin Mat Univ Padova, 124:231?C246, 2010) is: a finite solvable group G is a PST-group if and only if every subgroup of Fit(G) is ??-semipermutable. A local version of this result is established in this paper. A subgroup H of G is said to be ??-seminormal provided that it is normalized by all Sylow subgroups Q such that (|H|, |Q|) =?1 and (|H|, |Q ^G |) ???1. It is shown that a finite solvable group is a PST-group if and only if every subgroup of Fit(G) is ??-seminormal in G.     

On Weakly s-permutably Embedded Subgroups of Finite Groups

Suppose G is a finite group and H is subgroup of G. H is said to be s-permutably 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-permutable subgroup of G; H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup H _se of G contained in H such that G = HT and H ∩ T ≤ H _se . We investigate the influence of weakly s-permutably embedded subgroups on the structure of finite groups. Some recent results are generalized.     

Simple Twisted Group Algebras of Dimension p4 and their Semi-Centers

For a simple twisted group algebra over a group G, if G^∣ is Hall subgroup of G, then the semi-center is simple. Simple twisted group algebras correspond to groups of central type. We classify all groups of central type of order p⁴ where p is prime and use this to show that for odd primes p there exists a unique group G of order p⁴, such that there exists simple twisted group algebra over G with a commutative semi-center. Moreover, if 1 < |G| <64, then the semi-center of simple twisted group algebras over G is noncommutative and this bounds are strict.     

On weakly s-permutable subgroups of finite groups

A subgroup H of a group G is said to be weakly s-permutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ H _sG where H _sG is the largest s-quasinormal subgroup of G contained in H. In this paper, we investigate the influence of weak s-permutability of some primary subgroups in finite groups. Some new results about p-supersolvability and p-nilpotency of finite groups are obtained.     

Locally Finite Groups with All Subgroups Normal-by-(Finite Rank)

A group is said to have finite (special) rank ≤ sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/H_Ghas finite rank for all subgroupsHofG, whereH_Gdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/H_G, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3).     

Finite groups with given nearly s-embedded subgroups

Let G be a finite group and H a subgroup of G. We say that H is s-permutable in G if HP = PH for all Sylow subgroups P of G; H is s-semipermutable in G if HP = PH for all Sylow subgroups P of G with (|P|, |H|) = 1. Let H _s G be the subgroup of H generated by all those subgroups of G which are s-permutable in G and H ^sG the intersection of all such s-permutable subgroups of G contain H. We say that H is nearly s-embedded in G if G has an s-permutable subgroup T such that H ^sG = HT and \({H \cap T \leqq H_{ssG}}\) , where H _ssG is an s-semipermutable subgroup of G contained in H. In this paper, we study the structure of a finite group G under the assumption that some subgroups of prime power order are nearly s-embedded in G. A series of known results are improved and extended.     

s-Permutably embedded subgroups and p-supersolvable groups

Let G be a finite group,and H a subgroup of G.H is called s-permutably embedded in G if each Sylow subgroup of H is a Sylow subgroup of some s-permutable subgroup of G.In this paper,we use s-permutably embedding property of subgroups to characterize the p-supersolvability of finite groups,and obtain some interesting results which improve some recent results.     

The Influence of Weakly s-Supplemented Subgroups on the Structure of Finite Groups

Let G be a finite group. A subgroup H of G is said to be weakly s-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H _{s G} , where H _{s G} is the subgroup of H generated by all those subgroups of H which are s-quasinormal in G. In this article, we investigate the structure of G under the assumption that some families of subgroups of G are weakly s-supplemented in G. Some recent results are generalized.     

New characterizations of finite supersoluble groups

Let A be a subgroup of a group G and X a nonempty subset of G.A is called an X- semipermutable subgroup of G if A has a supplement T in G such that for every subgroup T_1 of T there exists an clement x∈X such that AT_i~x=T_i~xA.On the basis of this concept we obtain some new characterizations of finite supersoluble groups.     

Unit Groups of Integral Finite Group Rings with No Noncyclic Abelian Finite p-Subgroups

It is shown that in the units of augmentation one of an integral group ring ? G of a finite group G, a noncyclic subgroup of order p ², for some odd prime p, exists only if such a subgroup exists in G. The corresponding statement for p = 2 holds by the Brauer–Suzuki theorem, as recently observed by Kimmerle.     

Almost fixed-point-free automorphisms of prime order

Let ϕ be an automorphism of prime order p of the group G with C _G (ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are bounded in terms of p, n and h only. Here a group has finite Hirsch number if it is poly (cyclic or locally finite). This is a stronger notion than that used in [Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)], where the case p = 2 is discussed.     

c*-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-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; H is called c*-quasinormally embedded in G if there is a subgroup T of G such that G = HT and H??T is s-quasinormally embedded in G. We investigate the influence of c*-quasinormally embedded subgroups on the structure of finite groups. Some recent results are generalized.     

Finite groups with hall subnormally embedded Schmidt subgroups

Abstract

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded.