On Pronormal Minimal Subgroups of Finite Groups

By using pronormal minimal subgroups and weak left Engel elements ofprime order of the normalizers of Sylow subgroups of a finite group G, we obtain somesufficient conditions for G to be p-nilpotent, nilpotent and supersolvable respectively,which generalize some known results.

A Generalization of Cuccia＇s Theorem

For a finite group G, let S（G） be the set of minimal subgroups of odd order,which are complemented in G. It is proved that if every minimal subgroup X of odd orderof G which does not belong to S（G）, CG（X） is either subnormal or abnormal in G. Then Gsolvable.

CHARACTERIZATION OF MODULAR FROBENIUS GROUPS OF SPECIAL TYPE

In this article, we first investigate the properties of modular Frobenius groups. Then, we consider the case that G＇ is a minimal normal subgroup of a modular Frobenius group G. We give the complete classification of G when G＇ as a modular Frobenius kernel has no more than four conjugacy classes in G.

Finite p-Groups G with H'=G'for Each A2-Subgroup H^(*)

A finite p-group G is called an At-group if t is the minimal non-negative integer such that all subgroups of index pt of G are abelian.The finite p-groups G with H'=G'for all A2-subgroups H of G are classified completely in this paper.As an application,a problem proposed by Berkovich is solved.

Some open questions in the theory of generalized permutable subgroups

A subgroup H of a group G is said to be weakly s-supplemented in G if H has a supplement T in G such that H ∩ T HsG, where HsG is the largest s-permutable subgroup of G contained in H. This paper constructs an example to show that the open questions 6.3 and 6.4 in J Algebra, 315: 192–209 (2007) have negative solutions, and shows that in many cases Question 6.4 is positive. A series of known results are unified and generalized.

Semipermutable subgroups and s-semipermutable subgroups in finite groups

Suppose that H is a subgroup of a finite group G.We call H is semipermutable in G if HK=KH iov any subgroup K of G such that(|H|,|K|)=1;H is s-semipermutable in G if HG_(p)=G_(p)H,for any Sylow p-subgroup G_(p)of G such that(|H|,p)=1.These two concepts have been received the attention of many scholars in group theory since they were introduced by Professor Zhongmu Chen in 1987.In recent decades,there are a lot of papers published via the application of these concepts.Here we summarize the results in this area and gives some thoughts in the research process.

A question on partial CAP-subgroups of finite groups

A subgroup H of a finite group G is a partial CAP-subgroup of G if there is a chief series of G such that H either covers or avoids every chief factor of the series.The structural impact of the partial cover and avoidance property of some distinguished subgroups of a group has been studied by many authors.However,there are still some open questions which deserve an answer.The purpose of the present paper is to give a complete answer to one of these questions.

Thompson's conjecture for Lie type groups E_7(q)

In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type ET（q）. More precisely, we show that every finite group G with the properties Z（G） = 1 and cs（G） = cs（ET（q）） is necessarily isomorphic to ET（q）, where cs（G） and Z（G） are the set of lengths of conjugacy classes of G and the center of G respectively.