Finite Group with Two Supersolvable Subgroups of Coprime Indices

In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification theorems, we prove some sufficient conditions of finite solvable groups. Finally, we provide a supplement of Doerks Theorem.

Two Classes of Solvable Subgroups of SL（3, C） and Application for Differential Equations in C

Structures of two classes of solvable subgroups in SL（3, C） are given in this paper, and the integrability of the 3-order Fuchsian equation which is integrable in the sense that its monodromy group is solvable is discussed.

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.

Classification about Non-Solvable Groups with Exactly 40 Maximal Order Elements

Let φ be a homomorphism from a group H to a group Aut（N）. Denote by Hφ× N the semidirect product of N by H with homomorphism φ. This paper proves that： Let G be a finite nonsolvable group. If G has exactly 40 maximal order elements, then G is isomorphic to one of the following groups： （1） Z4φ×A5, kerφ = Z2; （2） D8φ ×A5, kerφ = Z2 ×Z2; （3） G/N = S5, N = Z（G） = Z2; （4） G/N = S5, N = Z2 ×Z2, N∩Z（G） = Z2.

On the Existence of p-Blocks with a Given p-Group as Defect Group G

In this paper, we study some actions of a finite group G on the set of characters of its subgroups, and by using these actions we determine the existence of a p-block with given defect group in some cases.

Some Sufficient Conditions of a Solvable Group

A subgroup H of a group G is called semipermutable if it is permutable with every subgroup K of G with （｜H｜, ｜K｜） = 1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with （p, ｜H｜） = 1. In this paper, some sufficient conditions for a group to be solvable are obtained in terms of s-semipermutability.

On Finite Groups with Special Conjugacy Classes

Let G be a finite group with the property that for any conjugacy class order, G has exactly two conjugacy classes which have the same order. We prove that： （1） ff a Sylow 2-subgroup of G is Abelian, then G is isomorphic to the direct product of symmetric group with order 3 and cyclic group with order 2, or G is isomorphic to the semidirect product of a cyclic group with order 3 and a cyclic group with order 4; （2） if G＇ is nilpotent, then G is a group of {2,3,5 }.

On s＊-Completions of Maximal Subgroups of Finite Groups

Given a maximal subgroup M of a group G, a θ*-completion C of M is called an s*-completion if either C = G or there exists a subgroup D of G which is not a θ*-completion of M such that D contains C as a maximal subgroup. In this paper, we obtain several results on s*-completions which imply G to be solvable or supersolvable.

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.

The Decomposition of Finite Groups

In this paper,we investigate the conditions under which a finite group can be decomposed,and present several sufficient conditions for it decompose.

A Sufficient Condition for Mi-groups

In this note, we give a sufficient condition for Mi-group. In particular, we show that if a finite group G is the semidirect product of two subgroups with coprime orders, in which one is a Sylow tower group and its Sylow subgroups are all abelian, and the other is an Mi-group and all of its proper subgroups are also Mi-groups, then G is an Mi-group.

Solvable line-transitive automorphism groups of finite linear spaces

Let S be a finite linear space, and let G be a group of automorphisms of S. If G is soluble and line-transitive, then for a given k but a finite number of pairs of ( S, G), S has v= pn points and G≤AΓ L(1,pn).

Some Sufficient Conditions on the Number of Non-abelian Subgroups of a Finite Group to be Solvable

Thompson＇s theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson＇s theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.

Solvable D_2-Groups

Let G be a finite group, Irra（G） be the set of nonlinear irreducible characters of G and cdl （G） the set of degrees of the characters in Irrl （G）. A group G is said to be a D2-group if │cdl （G）│ = │Irrl（G）│ - 2. In this paper, we give a complete classification of solvable D2-groups.

A characterization of finite simple groups by the orders of solvable subgroups

Let G be a finite group and S be a finite simple group. In this paper, we prove that if G and S have the same sets of all orders of solvable subgroups, then G is isomorphic to S, or G and S are isomorphic to Bn(q), Cn(q), where n≥3 and q is odd. This gives a positive answer to the problem put forward by Abe and Iiyori.

The Structure of One Type of Non-Abelian Group

By the property of the solvable group and the extending theorem of group, the authors acquired the structure of one type of Non-Abelian group. And we proved that when order is 10p n (p#2,5) and the sylowp-subgroup is cyclic, the group has twenty types. Whenp#3, it has 12 types and whenp=3, it has 8 types.

Regular and p-Regular Orbits of Solvable Linear Groups,Ⅱ

Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In this note,we study two questions,one by the authors and one by Isaacs,related to the p-regular orbits and regular orbits of the linear group actions.

MINIMAL SURFACES IN 3-DIMENSIONAL SOLVABLE LIE GROUPS

The author studies minimal surfaces in 3-dimensional solvable Lie groups with left invariantRiemannian metrics. A Weierstraβ type integral representation formula for minimal surfaces isobtained.

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).

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.