A note on solvability of finite groups

In this paper, we obtain the solvability for a finite group based on the assumption that Sylow 2-subgroups have the semi cover-avoiding properties. Some known results are generalized. Mathematics Subject Classification: 20D10, 20D20.

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Introduction

Gaschutz in [1] introduced pre-Frattini subgroups. These subgroups have cover-avoiding properties, that is, suppose that H  G, for any chief series 1= 0

that for every i=1, ,n either H covers

Many authors studies this property, for example, Ezquerro in [2] gave some characterization for a finite group G to be p-supersolvable and super solvable based on the assumption that all maximal subgroups of some Sylow subgroups of G have the cover-avoiding properties. Guo and Shum in [3] obtain some characterizations for a finite solvable group based on the assumption that some of its maximal subgroups or 2-maximal subgroups have the cover-avoiding properties. Recently, Fan, Guo and Shum in [4] generalized the cover-avoiding properties of finite groups. They called semi cover-avoiding properties, that is, a subgroup H is said to be semi coveravoiding in a group G if there is a chief series 1= 0

They used semi cover-avoiding properties of Sylow subgroups to investigate the solvability of finite groups. They proved the following result: Theorem A. [4] Let G be a finite group.Then G is solvable if and only if every Sylow subgroup of G is semi cover-avoiding subgroup of G. In this paper, we will go on to study the influence of semi cover-avoiding subgroups on the structure of finite groups. In Theorem A, we can replace every Sylow subgroup of G by every Sylow 2-subgroup of G, the Theorem still holds. Our main result is the following: Theorem B. Let G be a finite group.Then G is solvable if and only if every Sylow 2-subgroup of G is semi cover-avoiding subgroup of G. All groups considered in this paper are finite groups. Our notation and terminology are standard. The reader may refer to ref [5] .

Preliminary results

In this section, we give one lemma which are useful for our main results. Lemma 2.1 [6] Let G be a group. Let H be a semi cover-avoiding subgroup of G and N be a normal subgroup of G.

(i)If H  K  G, then H is a semi cover-avoiding subgroup of K. (ii)If N  H or gcd(|H|,|N|) = 1, where gcd(-, -) denotes the greatest common divisor, then HN/N is a semi cover-avoiding subgroup of G/N.

Proof of the Main Theorem

Proof of Theorem B: If G is solvable, then by Theorem A we know that every Sylow subgroup of G is semi cover-avoiding subgroup. So every Sylow 2-subgroup of G is semi coveravoiding subgroup. Conversely, let P be a Sylow 2-subgroup of G. If G is simple, then P covers or avoids the only chief series G/1. So G=P or 1    P P G . If G=P, then G is solvable. If P=1, then G is an group of odd order. By Feit-Thompson Theorem, we have that G is solvable. So we suppose that G is not simple and H cover or avoids a chief series 1= 0

If P N  , then P/N is a Sylow subgroup of G/N. By Lemma 2.1 we have that P/N is a semi cover-avoiding subgroup of G/N. By induction on |G| we know that G/N is solvable. Since P N  , N is solvable. So G is solvable. If N is not contained in P, then PN  P. So P  N=1. Since P is a Sylow 2subgroup of G and N  G, we have that P  N is a Sylow 2subgroup of N. So N is an odd group. That is, (|P|,|N|)=1. By Lemma 2.1 we have that PN/N is a Sylow 2-subgroup of G/N and is a semi cover-avoiding subgroup of G/N. So G/N satisfies the hypothesis of the theorem. By induction on |G|, we know that G/N is solvable. Since N is solvable, G is solvable. Our proof is complete now.

Acknowledgements

The paper is dedicated to Professor O.H.Kegel for his 80th birthday.

1. Gaschutz W. Praefrattin gruppen. Arch. Math. 1962; 13:418-426.
2. Ezquerro LM. A contribution to the theory of finite super solvable groups. Rend. Sem. Mat. Univ. Padova, 1993; 89:161-170.
3. Guo XY, Shum KP. Cover-avoidance property and the structure of finite groups. J.Pure Appl. Algebra. 2003; 181:297-308.
4. subgroups. Chinese Annals of Mathematics. 2006; 27A(2):169-176.
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6. Qiang Zhou, Hong Pan. On semi cover-avoiding subgroups of finite groups, International Journal of Statistics and Applied Mathematics. 2017; 2(2):41-42.