Research article · 2017 · Full text
International Journal of Statistics and Applied MathematicsOn semi cover-avoiding subgroups of finite groups
Volume 2(2) · Pages 41–42
Research areasFinite Group Theory Researchgraph theory and CDMA systemsSynthesis and properties of polymers
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.
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 cover-avoiding properties in finite groups began with the introduction of pre-Frattini subgroups, which either cover or avoid chief factors of a finite solvable group. View evidence ↗
- The concept of 'semi cover-avoiding properties' is a recent generalization of the original cover-avoiding properties of finite groups. View evidence ↗
- Previous research has utilized semi cover-avoiding properties of maximal and Sylow subgroups to study whether finite groups are solvable. View evidence ↗
The question
- The authors aim to determine if a finite group is supersolvable given that its minimal subgroups and cyclic subgroups of order 4 possess semi cover-avoiding properties. View evidence ↗
What they did
- The research focuses on proving Theorem A, which posits that if G/N is supersolvable and the minimal and order-4 cyclic subgroups of N are semi cover-avoiding in G, then G itself is supersolvable. View evidence ↗
- The proof of the main theorem is conducted using a proof by contradiction, assuming G is a counterexample of minimal order. View evidence ↗
What they found
- The authors establish that under the given conditions, a minimal counterexample G must be an inner-supersolvable group. View evidence ↗
- For an element x in P but not in Φ(P), the order of x is found to be either p or 4. View evidence ↗
- The authors demonstrate that G/Φ(P) is supersolvable, which implies G is supersolvable, contradicting the initial assumption that G was a counterexample. View evidence ↗
What it means
- The authors conclude that the supersolvability of a finite group G can be guaranteed if G/N is supersolvable and the minimal and order-4 cyclic subgroups of N are semi cover-avoiding in G. View evidence ↗
Visual summary
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Study overview
AI-generated conceptual illustration (Abstract mathematical group structures represented as geometric networks of nodes and edges). It explains the idea and does not show study data.
View source evidence ↗Introduction
All groups considered in this paper are finite groups. Our notation and terminology are standard. The reader may refer to ref [5] . In 1962, Gaschutz [1] introduced a certain conjugacy class of subgroups of a finite solvable group which he called pre-Frattini subgroups. These subgroups have the property that they not only avoid the complemented chief factors of a finite solvable group G but also cover the rest of its chief factors, we call these subgroups that have cover-avoiding properties, that is, suppose that H G, for any chief series 1= 0
Thereafter, 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 coveravoiding properties. Guo Xiuyun and K.P. Shum in [3] obtain some characterizations for a finite solvable group based on the assumption that some of its maximal subgroups or 2maximal subgroups have the cover-avoiding properties. Recently, in [4] , Fan, Guo and Shum 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. In this paper, we will push further this approach and 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. Our main result is the following: Theorem A. 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.
Basic definitions and preliminary results
In this section, we give one definition and some lemmas which are useful for our main results. Definition 2.1 ( [4] ) A subgroup H is said to be semi cover-avoiding in a group G if there is a chief series 1= 0
The following Lemma is obervous Lemma 2.2 Let H be a subgroup of a group G and let 1< <N< <M< <G be a normal series of G. If the subgroup H covers (respectively avoids) M/N, then H covers (respectively avoids) any quotient factor between N and M of any refinement of the normal series.
Lemma 2.3
Let G be a group. Let H be a semi cover-avoiding subgroup of G and N be a normal subgroup of G.
1. If H K G, then H is a semi cover-avoiding subgroup of K.
2. If N H or gcd(|H|,|N|) = 1, where gcd(-, -) denotes the greatest common divisor, then HN/N is a semi coveravoiding subgroup of G/N.
Proof. (i)Since H is a semi cover-avoiding subgroup of G, that is, there exists a chief series 1= 0
. So H semi covers or avoids a normal series 1=
. By Lemma 2.2 we know, H is a semi cover-avoiding subgroup of K. (ii) By Lemma 2.2 in [4] , we can prove the result. Lemma 2.4 ( [6] ) Suppose that G is an innersupersolvable group, then 1. there exists a normal PSly p (G) such that G=P M, P/ (P) is a minimal normal subgroup of G/ (P). (N). In fact, since G/P M and M is supersolvable, we have that G/P is also supersolvable. By hypothesis of the theorem, we know that G/N is supersolvable. So G/P N is supersolvable. If P N (P), then G/ (P) is supersolvable. So its quotient group G/ (G) is supersolvable, a contradiction. Thus P N can not be contained in (P). Since P/ (P) is a minimal normal subgroup of G/ (P), we have that (P N)\ (P)/ (P) is a normal p-subgroup of G/ (P). So P=(P N) (P)=(P N) p Syl (N).
Suppose that an element xP\ (P), then o(x)=p or o(x)=4. By hypothesis of the theorem, we know that x is a semi cover-avoiding subgroup of G. That is, there exists a chief series 1= 0
Acknowledgement
AcknowledgementsThe research is supported by the NNSF of China(11201400).The paper is dedicated to Professor John Cossy for his 75th birthday.
References
6 in the original paper- Gaschutz W. Praefrattin gruppen. Arch. Math. 1962; 13:418-426.
- Ezquerro LM. A contribution to the theory of finite supersolvable groups. Rend. Sem. Mat. Univ. Padova, 1993; 89:161-170.
- Guo XY, Shum KP. Cover-avoidance property and the structure of finite groups. J. Pure Appl. Algebra. 2003; 181:297-308.
- Fan Y, Guo XY, Shum KP. Remarks on two generalizations of normality of subgroups. Chinese Annals of Mathematics, 2006; 27A(2):169-176.
- Robinson DJS. A Course in the Theory of Groups. Spring-verlag, New York-Heidelberg-Berlin. 1980.
- Wang Pinchao, Wen Fengtong, Wexiang Li. C-normal subgroups of a finite group. Journal of Qufu Normal University, 1997; 23(4):5-7.
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- A note on solvability of finite groups
Meijun Guo, Qingjun Kong · International Journal of Statistics and Applied Mathematics · 2017
- A note on semi cover-avoiding subgroups of finite groups
Hong Pan, Qiang Zhou · International Journal of Statistics and Applied Mathematics · 2017
In 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