Research article · 2017 · Full text
International Journal of Statistics and Applied MathematicsA note on semi cover-avoiding subgroups of finite groups
Volume 2(4) · Pages 33–34
Research areasFinite Group Theory Researchgraph theory and CDMA systemsMatrix Theory and Algorithms
Abstract
Author-writtenIn this paper, we obtain the supersolvability for a finite group based on the assumption that minimal subgroups and cyclic subgroups of order 4 have the semi cover-avoiding properties. Some known results are generalized. Mathematics Subject Classification: 20D10, 20D20
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- The study of finite groups involves analyzing subgroups with cover-avoiding properties, a concept introduced by Gaschutz in 1962. View evidence ↗
- Semi cover-avoiding properties are a generalization of the cover-avoiding properties of finite groups. View evidence ↗
- Previous research established that a finite group is supersolvable if its minimal subgroups and cyclic subgroups of order 4 are semi cover-avoiding. View evidence ↗
The question
- The authors aim to further investigate how semi cover-avoiding subgroups influence the overall structure of finite groups. View evidence ↗
What they did
- The analysis is conducted using standard mathematical notation and terminology applied to finite groups. View evidence ↗
- The authors use a proof by contradiction, assuming the existence of a counterexample of minimal order to prove their main theorem. View evidence ↗
- The proof relies on a lemma stating that semi cover-avoiding properties are preserved in subgroups and under certain quotient conditions. View evidence ↗
What they found
- For a finite solvable group G where G/N is supersolvable, G is also supersolvable if every minimal subgroup and every cyclic subgroup of order 4 of the Fitting subgroup F(N) are semi cover-avoiding in G. View evidence ↗
- In the minimal counterexample analysis, the authors demonstrate that the Frattini subgroup (G) must be trivial (equal to 1). View evidence ↗
- The Fitting subgroup F(G) is found to be the direct product of minimal normal subgroups of G. View evidence ↗
What it means
- The authors conclude that the conditions specified in Theorem B are sufficient to prove that the group G is supersolvable. View evidence ↗
Visual summary
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Conceptual Hierarchy of Group Properties
AI-generated conceptual illustration (Abstract geometric representations of finite group structures). It explains the idea and does not show study data.
View source evidence ↗Introduction
In 1962, Gaschutz [1] introduced pre-Frattini subgroups. These subgroups have cover-avoiding properties, that is, suppose that H G, for any chief series 1= 0 G < 1 G << n G =G, such that for every i=1,,n either H covers i
Many authors studies this property, for example, In 1993, Ezquerro [2] gave some characterization for a finite group G to be p-supersolvable and supersolvable 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 [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 and maximal subgroups to investigate the solvability of finite groups. We [9] obtain the supersolvability for a finite group based on the assumption that minimal subgroups and cyclic subgroups of order 4 have the semi cover-avoiding properties. We proved the following result: Theorem A. [9] Let G be a finite group. N G and G/N is supersolvable. If every minimal subgroup and every cyclic subgroup of order 4 of N are semi cover-avoiding subgroups of G, then G is supersolvable. 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, if G is a finite solvable group, we can replace N by F(N). Our main result is the following: Theorem B. Let G be a finite solvable group. N G and G/N is supersolvable. If every minimal subgroup and every cyclic subgroup of order 4 of F(N) are semi cover-avoiding subgroups of G, then G is supersolvable. 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 [9] Let G be a group. Let H be a semi coveravoiding 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. Assume that the theorem is false and let G be a counterexample of minimal order. Then
In fact, since G is solvable,by [6,III,4.2,c], we have that F(G)/ (G)>1. If (G)>1, we can suppose that K is a minimal normal subgroup of G such that K (G), then F(G/K)=F(G)/K. By lemma 2.3, we know that G/K satisfies the hypothesis of the theorem, so G/K is supersolvable. Thus G is supersolvable. a contradiction, so (G)=1.
, where j N (j=1,2,,s) and
In fact, by [7, Lemma 2.3 of Appendixes] we know that F (G) is the direct product of minimal normal subgroups of G. By the Lemma 2.3 in [8] , we know F
, where j N (j=1,2,s) are minimal normal subgroups of G. Noticing that F(N) F(G), we know (ii) is true.
(iii)Every j N (j=1,2,,s) and j M (j=1,2,,t) are cyclic groups of prime order. In fact, by Theorem 3.2 in [4] , we know that F(N) is supersolvable, so every j N (j=1,2,,s) is a cyclic group of prime order. Since N N M j / is a minimal normal subgroup of G/N, we have that N N M j / is a cyclic group of prime order by the supersolvability of G/N. Hence
No matter what case it is, we have that [F(G): M F(G)] is a prime. By [8,p12 Theorem 3.3], G is supersolvable. Our proof is complete now.
Acknowledgements
The research is supported by the NNSF of China (11201400).The paper is dedicated to Professor John Cossy for his 75th birthday.
References
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