Finite groups with seminormal Schmidt subgroups

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB₁ is a proper subgroup of G, for every proper subgroup B₁ of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary. Supported by BelFBR grant Nos. F05-341 and F06MS-017. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.     

两个正规p-超可解子群的积

本文研究了两个正规的p-超可解子群的积构成的极小非P-超可解群的结构的问题.利用有限群论和群类论的一些基本方法,获得了两个正规的p-超可解子群的积仍为P-超可解群的一些充分条件,并推广了文[1]中关于超可解群的情况.     

Finite groups with a seminormal Hall subgroup

Tests for π-solvability of a finite group with seminormal Hall π-subgroup are established and the nilpotency of the third commutator subgroup of any group with seminormal noncyclic Sylow subgroups is proved.     

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 complemented subgroups of finite groups

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H ⋂ K = 1. In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about p-nilpotent groups.     

Local embeddings of some families of subgroups of finite groups

In this paper we study the influence of the partial cover and avoidance property on the subgroups of some relevant families of subgroups in a finite group.     

The influence of X-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.     

Fuzzy subgroups of nilpotent groups

In 1991, Asaad has suggested the following question: What can be said about the structure of a group G when information is known about the structure of a fuzzy subgroup A of G and vice-versa? This question has been partially investigated in Fuzzy Sets and Systems39 (1991) 323–328. The purpose of this paper is to continue more investigations for the above mentioned question.     

<Emphasis Type="Italic">c</Emphasis>-Semipermutable subgroups of finite groups

A subgroup is called c-semipermutable in G if A has a minimal supplement T in G such that for every subgroup T ₁ of T there is an element x ∈ T satisfying AT ₁ ^x = T ₁ ^x A. We obtain a few results about the c-semipermutable subgroups and use them to determine the structures of some finite groups.     

On the existence of solvable normal subgroups in finite groups

The structure of a finite group in dependence on the structure of the subgroups generated by elements of its conjugate class is considered. Translated fromMatematischeskie Zametki, Vol. 61, No. 5, pp. 755–758, May, 1997. Translated by A. I. Shtern     

Nearly m-embedded subgroups and solvability of finite groups

A subgroup A of a finite group G is called {1≤G}-embedded in G if for each two subgroups K≤H of G, where K is a maximal subgroup of H, A either covers the pair (K,H) or avoids it. Moreover, a subgroup H of G is called nearly m-embedded in G if G has a subgroup T and a {1≤G}-embedded subgroup C such that G?=?HT and H∩T≤C≤H. In this paper, we mainly prove that G is solvable if and only if its Sylow 3-subgroups, Sylow 5-subgroups and Sylow 7-subgroups are nearly m-embedded in G.     

G-Covering Systems of Subgroups for Classes of p-Supersoluble and p-Nilpotent Finite Groups

Let be a class of groups. Given a group G, assign to G some set of its subgroups =(G). We say that is a G-covering system of subgroups for (or, in other words, an -covering system of subgroups in G) if G whenever either =Ø or Ø and every subgroup in belongs to . We find the systems of subgroups in the class of finite soluble groups G which are simultaneously the G-covering systems of subgroups for the classes of p-supersoluble and p-nilpotent groups.     

On the regular Sylow p-subgroups of Chevalley groups over \mathbb{Z}_{p^m }

We prove that a Sylow p-subgroup of the general linear group of dimension n over the residue ring modulo p ^m is regular for n ² < p and powerful if and only if n = 2 and m = 1. We obtain similar results for the Sylow p-subgroups of normal types over the same ring.     

Finite groups with some pronormal subgroups

A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H ^x in 〈H, H ^x 〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.     

Nonsoluble length of finite groups with restrictions on Sylow subgroups

By 𝔛(n) we denote the variety of all groups satisfying the law [x,y]ⁿ≡1, that is, groups with commutators of order dividing n. Let p be a prime and G a finite group whose Sylow p-subgroups have normal series of length k all of whose quotients belong to 𝔛(n). We show that the non-p-soluble length λ_p(G) of G is bounded in terms of k and n only (Theorem 1.2). In the case where p is odd, a stronger result is obtained (Theorem 1.3).     

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.     

Finite Sylow Subgroups in Simple Locally Finite Groups of Lie Type

The main result of the paper is the followingTheorem. Let S = {r₀, r₁, ..., r_n} be a finite nonempty set of primes and let L be a Lie type of Chevalley groups. Then there exists a locally finite field F of characteristic r₀ such that Sylow r-subgroups of the simple group L(F) of type L over F are finite if and only if r ? S.     

On weakly S-embedded subgroups of finite groups

Let H be a subgroup of a group G.Then H is said to be S-quasinormal in G if HP = P H for every Sylow subgroup P of G;H is said to be S-quasinormally embedded in G if a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G for each prime p dividing the order of H.In this paper,we say that H is weakly S-embedded in G if G has a normal subgroup T such that HT is an S-quasinormal subgroup of G and H ∩ T≤H SE,where H SE denotes the subgroup of H generated by all those subgroups of ...