Finite Groups in Which Every Maximal Subgroup Is Nilpotent or a TI-Subgroup or Has Order p'

We obtain a complete characterization of the structure of a finite group G in which every maximal subgroup is nilpotent or a TI-subgroup or has order p'for any fixed prime divisor p of|G|.Moreover,we show that there exists at most one prime divisor q of|G|such that G is neither q-nilpotent nor q-closed.

On-Abnormal Maximal Subgroups of Finite Groups

For any saturated formation F of finite groups containing all supersolvable groups, the groups in F are characterized by F-abnormal maximal subgroups.

Maximal von Neumann subalgebras arising from maximal subgroups

Ge(2003)asked the question whether LF∞can be embedded into LF2 as a maximal subfactor.We answer it affirmatively with three different approaches,all containing the same key ingredient:the existence of maximal subgroups with infinite index.We also show that point stabilizer subgroups for every faithful,4-transitive action on an infinite set give rise to maximal von Neumann subalgebras.By combining this with the known results on constructing faithful,highly transitive actions,we get many maximal von Neumann subalgebras arising from maximal subgroups with infinite index.

Maximal Subgroups in the Classical Groups Normalizing Solvable Subgroups

Let G be a classical group over an arbitrary field F,acting on an n-dimensional vector space V=V(n,F)over a field F.In this paper,we classify the maximal subgroups of G,which normalizes a solvable subgroup N of GL(L,F)not lying in F^(*)1_(V).

Existence of Nonabelian Free Subgroups in the Maximal Subgroups of GLn（D）

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of CLn （D） contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of Mn（D） such that NCLn（D）（K＊） = M, K＊ △ M, K/F is Galois with Gal（K/F） ≌ M/K＊, and F[M] = in（D）. In particular, when F is global or local, it is proved that if （[D ： F], Char（F）） = 1, then every non- abelian maximal subgroup of GL1 （D） contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn（F） contains no solvable maximal subgroups provided that F is local or global and n ≥ 5.

Maximal Subgroups in the Classical Groups Normalizing Solvable Subgroups

Let G be a classical group over an arbitrary field F,acting on an n-dimensional F-space V=V(n,F).All those maximal subgroups of G are classified each of which normalizes a solvable subgroup N of GL(V/F)not lying in F^(*)1v.

The Intersection of a Maximal Intransitive Subgroup with a Maximal Imprimitive Subgroup

In this paper, we investegate the intersection of a maximal intransitive subgroup with a maximal imprimitive subgroup. And, the structure of the second maximal intransitive subgroup of an alternating group is determined.

The Applications of Maximal Subgroup of Flute Group

For a maximal subgroup M of a finite group G, the normal index of M is defined to be the order of a chief factor H/K, where H is minimal supplement of M in G. For A【G , if there are two normal subgroup L and J of G such that G = A·L and A∩L= J, we say that A is an almost normal subgroup in G. We obtain several results that G to be solvable and supersolvable.

关于F-反正规子群的θ*-完备（英文）

Let F be a saturated formation of finite groups. Given a proper subgroup H of a group G, a subgroup H of G is called F-normal in G if G/HGbelongs to F; otherwise H is said to be F-abnormal in G. In this paper, we investigate the structure of a finite group by θ*-completions of F-abnormal subgroups.

Sub-cover-avoidance Properties and the Structure of Finite Groups

A subgroup H of a group G is said to have the sub-cover-avoidance property in G ffthereis a chief series 1 = G0 ≤ G1 ≤…≤ Gn - G, such that Gi-1（H ∩ Gi） G for every i = 1,2,... ,l. In this paper, we give some characteristic conditions for a group to be solvable under the assumptions that some subgroups of a group satisfy the sub-cover-avoidance property.

A New Characterization of Simple K_3-groups

In this paper, We show that the simple K\-3-groups can be characterized by the orders of their maximal abelian subgroups. That is, we have Theorem Let G be a finite group and M a simple K \-3-group. Then G is isomorphic to M if and only if the set of the orders of the maximal abelian subgoups of G is the same as that of M .

On p-CAP-subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G.H is said to be a p-CAP-subgroup of G if H either covers or avoids each pd-chief factor of G.We give some characterizations for a group G to be p-solvable under the assumption that some subgroups of G are p-CAP-subgroups of G.

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 Criterion for Subnormality of Subgroups in Finite Groups

Based on Wielandt＇s criterion for subnormality of subgroups in finite groups, we study 2-maximal subgroups of finite groups and present another subnormality criterion in finite solvable groups.

Finite NPDM-groups

In this paper the classification is given for finite groups in which the normalizer of every non-normal cyclic subgroup of order divided by the minimal prime of |G| is a maximal subgroup.

On Two Theorems of Finite Solvable Groups

For a finite group G, let T（G） denote a set of primes such that a prime p belongs to T（G） if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions： （1） G has a p-complement for each p∈T（G）; （2）│T（G）│= 2： （3） the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈T（G）; then G either is solvable or G/Sol（G）≌PSL（2, 7） where Sol（G） is the largest solvable normal subgroup of G.

Semi-Cover-Avoiding Properties and the Structure of Finite Groups

In this paper, the so-called π-cover-avoiding properties of subgroups are defined and investigated. In terms of this property, we characterize the π-solvability of finite groups. Some other new results are also obtained based on the assumption that some subgroups have the semi cover-avoiding properties in a finite group.