Nonsolvable Groups with Three Nonlinear Irreducible Character Codegrees

For an irreducible characterχof a?nite group G,the codegree ofχis de-?ned as|G:ker(χ)|/χ(1).In this paper,the authors determine?nite nonsolvable groups with exactly three nonlinear irreducible character codegrees,which are L_(2)(2^(f))for f≥2,PGL_(2)(q)for odd q≥5 or M_(10).

Nonsolvable D_2-groups

Let G be a finite group. Let Irrl（G） be the set of nonlinear irreducible characters of G and cdl（G） the set of degrees of the characters in Irr1（G）. A group G is said to be a D2-group if led1 （G）[ = ]Irr1 （G）I - 2. The main purpose of this paper is to classify nonsolvable D2-groups. Keywords Character degree, degree multiplicity, nonsolvable group

Nonsolvable groups whose irreducible character degrees have special 2-parts

Let G be a nonsolvable group and Irr(G)the set of irreducible complex characters of G.We consider the nonsolvable groups whose character degrees have special 2-parts and prove that ifχ(1)_(2)=1 or∣G∣_(2)for everyχ∈Irr(G),then there exists a minimal normal subgroup N of G such that N≅PSL(2,2^(n))and G/N is an odd order group.

The Characterizations of A_5 and PSL(2,7) and a Note on Finite CIT-Groups

In this paper, it has been shown that A_5 and PSL (2,7) are the only simple groups which just contain 15 or 21 in volutions, respectively. Meanwhile, we have also obtained some results about finite CIT-groups and finite nonsolvabie CIT-groups.

Study on Immobilized Lipase Catalyzed Transesterification Reaction of Tung Oil

The transesterification reaction conditions of tung oil with methanol have been studied in this article, with immobilized lipase NOVO435 as catalyst. The response surface methodology was used to optimize the transesterification reaction of tung oil in a nonsolvent system. The optimal conditions were rotation rate 200 r/min, molar ratio of methanol to oil 2.2： l, reaction temperature 43℃, and the catalyst amount 14% （based on the weight of oil）. After reacting for 18 h, 67.5% of the oil was converted to its corresponding methyl esters （the theoretical ester conversion was 73.3%）. The lipase was washed by organic solvents after each reaction and was reused again. The esters conversion of tung oil was decreased by 6% after the lipase was reused for 120 h. The theoretical amount of methanol was added in two steps, 85% ester conversion was obtained after 36 h of reaction （theoretical ester conversion was 100%）. The molar ratio of methanol to oil, the catalyst amount, the reaction temperature, and reaction time were all highly significant factors, and there was a relative significant interaction between every two factors.

A quantitative characterization of finite simple groups

In this note, all groups are finite and the notations and termentary are standard, inaddition, given a finite G group. let π(G) and π(|G|) denote the set of primes dividing |G|.

On Covering Number of Groups with Trivial Fitting Subgroup

Let G be a finite group and S be a subset of Irr(G).If for every prime divisor p of|G|there is a characterχin S such that p dividesχ(1),S is called a covering set of G.The covering number of G,denoted by cn(G),is defined as the minimal number of Card(S),where S is a covering set of G and Card(S)is the cardinality of set S.In this paper,we prove that if G is a finite group with F(G)=1,then the covering number cn(G)≤3.Especially,if PSL2(q)or J1 is not involved in G,then cn(G)≤2.

Bicontinuous porous membranes with micro-nano composite structure using a facile atomization-assisted nonsolvent induced phase separation method

The micro-nano composite structure can endow separation membranes with special surface properties,but it often has the problems of inefficient preparation process and poor structural stability.In this work,a novel atomization-assisted nonsolvent induced phase separation method,which is also highly efficient and very simple,has been developed.By using this method,a bicontinuous porous microfiltration membrane with robust micro-nano composite structure was obtained via commercially available polymers of polyacrylonitrile and polyvinylpyrrolidone.The formation mechanism of the micro-nano composite structure was proposed.The microphase separation of polyacrylonitrile and polyvinylpyrrolidone components during the atomization pretreatment process and the hydrogen bonding between polyacrylonitrile and polyvinylpyrrolidone molecules should have resulted in the nano-protrusions on the membrane skeleton.The membrane exhibits superhydrophilicity in air and superoleophobicity underwater.The membrane can separate both surfactant-free and surfactant-stabilized oil-in-water emulsions with high separation efficiency and permeation flux.With excellent antifouling property and robust microstructure,the membrane can easily be recycled for long-term separation.Furthermore,the scale-up verification from laboratory preparation to continuous production has been achieved.The simple,efficient,cost-effective preparation method and excellent membrane properties indicate the great potential of the developed membranes in practical applications.