On subgroups which cover or avoid chief factors of a finite group

A classical topic of research in Finite Group Theory is the following: What is the influence on the structure of the group of the fact that all members of some relevant family of subgoups enjoy a given embedding property? The cover-avoidance property is a subgroup embedding property that has r...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2009
Hauptverfasser:	Ballester-Bolinches, A., Ezquerro, L.M., Skiba, A.N.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2009
Schriftenreihe:	Algebra and Discrete Mathematics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/154564
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	On subgroups which cover or avoid chief factors of a finite group / A. Ballester-Bolinches, L.M. Ezquerro, A.N. Skiba // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 18–28. — Бібліогр.: 28 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-154564
record_format	dspace
spelling	irk-123456789-1545642019-08-31T22:40:25Z On subgroups which cover or avoid chief factors of a finite group Ballester-Bolinches, A. Ezquerro, L.M. Skiba, A.N. A classical topic of research in Finite Group Theory is the following: What is the influence on the structure of the group of the fact that all members of some relevant family of subgoups enjoy a given embedding property? The cover-avoidance property is a subgroup embedding property that has recovered much attention in the last few years. In this survey article we present a number of results showing that its influence goes much further from the classical upersolubility. 2009 Article On subgroups which cover or avoid chief factors of a finite group / A. Ballester-Bolinches, L.M. Ezquerro, A.N. Skiba // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 18–28. — Бібліогр.: 28 назв. — англ. 1726-3255 000 Mathematics Subject Classification:20D10, 20D15, 20D20. http://dspace.nbuv.gov.ua/handle/123456789/154564 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A classical topic of research in Finite Group Theory is the following: What is the influence on the structure of the group of the fact that all members of some relevant family of subgoups enjoy a given embedding property? The cover-avoidance property is a subgroup embedding property that has recovered much attention in the last few years. In this survey article we present a number of results showing that its influence goes much further from the classical upersolubility.
format	Article
author	Ballester-Bolinches, A. Ezquerro, L.M. Skiba, A.N.
spellingShingle	Ballester-Bolinches, A. Ezquerro, L.M. Skiba, A.N. On subgroups which cover or avoid chief factors of a finite group Algebra and Discrete Mathematics
author_facet	Ballester-Bolinches, A. Ezquerro, L.M. Skiba, A.N.
author_sort	Ballester-Bolinches, A.
title	On subgroups which cover or avoid chief factors of a finite group
title_short	On subgroups which cover or avoid chief factors of a finite group
title_full	On subgroups which cover or avoid chief factors of a finite group
title_fullStr	On subgroups which cover or avoid chief factors of a finite group
title_full_unstemmed	On subgroups which cover or avoid chief factors of a finite group
title_sort	on subgroups which cover or avoid chief factors of a finite group
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/154564
citation_txt	On subgroups which cover or avoid chief factors of a finite group / A. Ballester-Bolinches, L.M. Ezquerro, A.N. Skiba // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 18–28. — Бібліогр.: 28 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT ballesterbolinchesa onsubgroupswhichcoveroravoidchieffactorsofafinitegroup AT ezquerrolm onsubgroupswhichcoveroravoidchieffactorsofafinitegroup AT skibaan onsubgroupswhichcoveroravoidchieffactorsofafinitegroup
first_indexed	2025-07-14T06:37:29Z
last_indexed	2025-07-14T06:37:29Z
_version_	1837603278391607296
fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics SURVEY ARTICLE Number 4. (2009). pp. 18 – 28 c⃝ Journal “Algebra and Discrete Mathematics” On subgroups which cover or avoid chief factors of a finite group A. Ballester-Bolinches, Luis M. Ezquerro and Alexander N. Skiba Communicated by I. Ya. Subbotin Dedicated to Professor Leonid A. Kurdachenko on the occasion of his 60th birthday Abstract. A classical topic of research in Finite Group Theory is the following: What is the influence on the structure of the group of the fact that all members of some relevant family of subgoups enjoy a given embedding property? The cover-avoidance property is a subgroup embedding prop- erty that has recovered much attention in the last few years. In this survey article we present a number of results showing that its influence goes much further from the classical supersolubility. 1. Introduction In this survey article all groups are assumed to be finite. If a subgroup A of a group G has the property that either HA = KA or A∩H = A∩K for every chief factor H/K of G, then A is said to have the cover-avoidance The research of the first author is supported by Proyecto MTM2007-68010-C03-02, Ministerio de Educación y Ciencia de España. The research of the second author is supported by Proyecto MTM2007-68010-C03-01, Ministerio de Educación y Ciencia de España. 2000 Mathematics Subject Classification: 20D10, 20D15, 20D20. Key words and phrases: Finite group; cover-avoidance property; p-supersoluble groups; saturated formation. Jo u rn al A lg eb ra D is cr et e M at h .A. Ballester-Bolinches, L. M. Ezquerro, A. N. Skiba 19 property in G and is called a CAP-subgroup of G. This subgroup em- bedding property has afforded the attention of many authors. Some of them were interested in discovering some distinguished families of CAP- subgroups, mainly in the soluble universe, while others discovered some characterisations of soluble and supersoluble groups, or their correspond- ing local versions, in terms of CAP-property of the members of some relevant familes of subgroups. The reader may consult [2, Chapter 4], [3], [4], [15] and [28]. The purpose of this survey is to present some recent results about how the CAP-property of some distinguished subgroups of a group influences in its structure. In this context, the natural starting point is to think on groups in which every subgroup has the CAP-property. These groups are exactly the supersoluble ones. Recall that if p is a prime, a group G is said to be p-supersoluble if chief factor of G is cyclic of order p or a p′-group. The group G is supersoluble if it is p-supersoluble for all primes p, i.e., every chief factor of G is a cyclic group of prime order. Our study confirms that in many cases the cover and avoidance prop- erty of some restricted families of subgroups gives rise to structural results involving saturated formations containing the class of all supersoluble groups. Recall that a class of groups F is said to be a formation if F is closed under taking epimorphic images and subdirect products. F is saturated if is closed under taking Frattini extensions. It is well-known that the class of all supersoluble groups is a saturated formation Three of the most popular families in the CAP context are the ones of maximal, 2-maximal and minimal subgroups of the Sylow subgroups. The following family includes them as particular cases: Let p be a prime. For a group G and a p-subgroup D of G, we define: (1) If p is odd, then ℱD(G) = {H ≤ G : ∣H∣ = ∣D∣} (2) If p = 2, then ℱD(G) = {H ≤ G : ∣H∣ ∈ {∣D∣, 2∣D∣}} 2. Strong CAP-subgroups Unfortunately, the cover-avoidance property is not inherited in interme- diate subgroups. This means that if A is a CAP-subgroup of G and A is a subgroup of B, then A is not, in general, a CAP-subgroup of B. Example 2.1. ([3, Example 1.3]) Let G = Sym(6) be the symmetric group of degree 6. Consider the following subgroups of G: A = ⟨(123), (12)(34), (14)(23)⟩ ∼= Alt(4), C = ⟨(56)⟩, P = ⟨(12)(34), (14)(23), (56)⟩ ∼= C2 × C2 × C2 and H = Jo u rn al A lg eb ra D is cr et e M at h .20 On the cover-avoidance property ⟨(12)(34)(56)⟩. Since C ∩ A = 1 and C ≤ CG(A), take K = A × C. Then, G = HN , for N = Alt(6) and H is a CAP-subgroup of G. However, P/C is a chief factor of K and C < HC = ⟨(12)(34), (56)⟩ < P . Then, H is not a CAP-subgroup of K. Definition 2.2. Let A be a subgroup of a group G. Then we say that A is a strong CAP -subgroup of G if A is a CAP -subgroup of any subgroup of G containing A. Our main result of this section anaylises the impact of the strong CAP-property of some subgroups of the generalised Fitting subgroup on the structure of the group. Theorem 2.3. ([4, Theorem A]) Let F be a saturated formation con- taining all supersoluble groups and G a group with a normal subgroup E such that G/E ∈ F. Suppose that every non-cyclic Sylow subgroup P of the generalised Fitting subgroup F ∗(E) of E has a subgroup D such that 1 < ∣D∣ < ∣P ∣ and all subgroups H ∈ ℱD(P ) are strong CAP-subgroups of G. Then G ∈ F. For the saturated formation of all supersoluble groups we have: Corollary 2.4. ([4, Corollary 2]) A group G is supersoluble if and only if every non-cyclic Sylow subgroup P of F ∗(G) has a subgroup D such that 1 < ∣D∣ < ∣P ∣ and all subgroups H of P with order ∣H∣ = ∣D∣ and with order 2∣D∣ (if P is a non-abelian 2-group) are strong CAP-subgroups of G. 3. Partial CAP-subgroups Another subgroup embedding property in the CAP orbit is the following. Definition 3.1. Let A be a subgroup of a group G. Then, we say that A has the partial cover and avoidance property or A is a partial CAP- subgroup (or semi CAP-subgroup) of G if there exists a chief series ΓA of G such that A either covers or avoids each factor of ΓA. This embedding property was first studied by Y. Fan, X. Guo and K. P. Shum in [11]. They called these subgroups semi CAP-subgroups. This is the name used in the subsequent papers [14], [15], [17], [7] and [28]. Nevertheless, since in this case the prefix "semi" does not refer to any half, we think that the name partial CAP-subgroup is more descriptive. It is clear that CAP-subgroups are partial CAP-subgroups, but the converse does not hold in general. Jo u rn al A lg eb ra D is cr et e M at h .A. Ballester-Bolinches, L. M. Ezquerro, A. N. Skiba 21 Example 3.2. Consider as in Example 2.1 the symmetric group of degree 6, G = Sym(6). The subgroup H is a partial CAP-subgroup of K but H is not a CAP-subgroup of K. In the chief series of K: 1 < V < P < K H ∩ V = 1, i.e. H avoids V/1, P = HV , i.e. H covers P/V and H trivially avoids K/P , since H ≤ P . The partial cover and avoidance property is also an extension of the c-normality introduced by Wang in [27]. However it is rather easy to construct examples showing that the partial cover and avoidance property does not imply c-normality. Our study of the partial CAP-property tries to answer the following question: Let G be a group. If all members of the family ℱ of subgroups of G are partial CAP-subgroups of G, then what is the structure of G?. The family ℱ will be one of the following. (1) ℱ : The meet-irreducible subgroups of G with order divisible by p. (2) ℱ : The meet-irreducible subgroups with order a multiple of p of each maximal subgroup of G. (3) ℱ : The Sylow p-subgroups of G. (4) ℱ : The maximal subgroups of the Sylow p-subgroups of G. (5) ℱ : The 2-maximal subgroups of the Sylow p-subgroups of G. 3.1. On meet-irreducible subgroups. The following definition play a central role in this subsection. Definition 3.3. (D. L. Johnson. [24]) Let G be a group. A subgroup H is said to be meet-irreducible in G if whenever H = X1 ∩ . . . ∩ Xn, for some subgroups X1, . . . , Xn of G, then H = Xi for some i. This is clearly equivalent to say that H is a proper subgroup of the intersection of all subgroups of G which properly contain H. A word on terminology: D. L. Johnson names these subgroups primi- tive and observes that they are also called “meet-irreducible.” We prefer this last name to avoid any confusion with the classical primitive groups. Jo u rn al A lg eb ra D is cr et e M at h .22 On the cover-avoidance property The following result gives a characterisation of the p-supersoluble groups in terms of families of meet-irreducible subgroups whose order is divisible by p. Theorem 3.4. ([3, Theorem 3.10]) Let G be a group and p a prime dividing the order of G. The following conditions are pairwise equivalent: (1) G is a p-supersoluble group. (2) Every meet-irreducible subgroup of G with order divisible by p is a partial CAP-subgroup of G. (3) Every meet-irreducible subgroup with order a multiple of p of each maximal subgroup of G is a partial CAP-subgroup of G. 3.2. On Sylow p-subgroups and its maximal subgroups. Next we fix a prime p and analyse the structure of a group in which the members of the families (3), (4) and (5) above are partial CAP-subgroups. Y. Fan,X. Guo and K. P. Shum proved in [11] the following facts: ∙ The p-solubility of a group can be characterised by the partial CAP- property of the Sylow p-subgroups. ∙ The supersolubility of a group is characterised by the partial CAP- property of the maximal subgroups of the Sylow subgroup of the group. The method we used here is a local one. This means that it is gener- alised in a form referring to a prime. The reason for choosing this local method is to discover new situations. Roughly speaking, the global hy- pothesis, referring to all primes, force the solubility and some non-soluble cases do not appear. This also leads us to the interesting question of how the global properties can be obtained as the conjunction of the local ones for all primes. Theorem 3.5. ([3, Theorem 3.2]) Let p be a prime dividing the order of a group G. Then, all maximal subgroups of every Sylow p-subgroup of G are partial CAP-subgroups of G if and only if (1) either G is a group whose Sylow p-subgroups are cyclic groups of order p (2) or G is a p-supersoluble group. Jo u rn al A lg eb ra D is cr et e M at h .A. Ballester-Bolinches, L. M. Ezquerro, A. N. Skiba 23 Corollary 3.6. ([11]) Suppose that all maximal subgroups of every Sylow subgroup of a group G are partial CAP-subgroups of G. Then G is a supersoluble group. Theorem 3.7. ([3, Corollary 3.7]) Let G be a finite group and p a prime dividing the order of G. If every maximal subgroup of every Sylow p- subgroup of F ∗ p (G) is a partial CAP-subgroup of G, then either (1) G is a p-supersoluble group, or (2) F ∗ p (G)/Op′(G) is isomorphic to a non-abelian simple group whose Sylow p-subgroups are cyclic groups of order p. Here, F ∗ p (G) denotes the characteristic subgroup F∗ p(G) = ∩ {HCG(H/K) : H/K is a chief factor of G and p divides ∣H/K∣}. 3.3. The next aim Our next aim is the characterisation of the class of all groups G enjoying the following local property: (†) Every 2-maximal subgroup of every Sylow p-subgroup of G is a partial CAP-subgroup of G. Definition 3.8. Given a group G, a subgroup K of G is called a second maximal subgroup, or 2-maximal subgroup, if there exists a maximal subgroup M of G such that K ≤ M and K is maximal in M . If G is a p-group (p a prime) of order pn, then the 2-maximal sub- groups of G are all subgroups of order pn−2. The idea of obtaining information on the structure of a group in which some 2-maximal subgroups are embedded in some particular way has produced many results. Let us emumerate some of them. (1) B. Huppert [21] prove the following: If every 2-maximal subgroup of a group G is normal in G, then G is supersoluble, and if moreover the order of G is divisible by at least three distinct primes, then G is nilpotent. (2) A. Mann [25] obtained information on the structure of all groups whose 2-maximal subgroups are subnormal. Jo u rn al A lg eb ra D is cr et e M at h .24 On the cover-avoidance property (3) R. K. Agrawal [1] obtained Huppert’s result under hypothesis of permutability. (4) Li Shirong [26] obtained the structure of all groups whose 2-maximal subgroups are TI-subgroups. (H is a TI-subgroup of G, if for every g ∈ G, we have H ∩Hg = 1.) (5) Guo, Shum and Skiba [16] analysed some conditions of supersolubil- ity and nilpotency of groups by means of the X-semiper-mutability of either 2-maximal subgroups or maximal subgroups. In our study we must have some important facts in mind. ∙ The trivial case: If the Sylow p-subgroups have order p2, then the family of all 2-maximal subgroups of the Sylow p-subgroups is com- posed of just the trivial subgroup. Trivially these groups verify property (†). Hence, what we will call “the trivial case” will be the characterisation of all groups whose Sylow p-subgroups have order p2. ∙ By the Brauer-Suzuki Theorem, if the Sylow 2-subgroups of a group G are isomorphic to Q8, then Z(G/O2′(G)) has even order. Hence every group whose Sylow 2-subgroups are isomorphic to Q8 has property (†). We shall include these groups in our characterisation. ∙ The p-soluble case: The class of all p-supersoluble groups is com- posed of groups with property (†). Thus, our interest will be to characterise all p-soluble groups with property (†) which are not p-supersoluble. This is the most complicated part of the question. The trivial case. Theorem 3.9. ([5]) Let G be a group with Sylow p-subgroups of order p2. Let us denote G = G/Op′(G) S = Soc(G) F = Op′(G). Then we have one of the following. (1) Either S is the direct product of two distinct minimal normal sub- groups of G, say N1 and N2. In this case, F = S and N1 and N2 are simple groups with cyclic Sylow p-subgroups of order p. (2) Or S is a chief factor of G. In this case we have one of the following. Jo u rn al A lg eb ra D is cr et e M at h .A. Ballester-Bolinches, L. M. Ezquerro, A. N. Skiba 25 (a) S ∼= Cp; in this case G is p-supersoluble and F ∼= Cp2 . (b) G is a primitive group of type 1; in this case G is p-soluble and F = S ∼= Cp × Cp. (c) G is a primitive group of type 2 and F = S; in this case, ∙ either S ∼= T × T , T non-abelian simple with Sylow p- subgroups of order p, ∙ or S non-abelian simple with Sylow p-subgroups of order p2. (d) G is a primitive group of type 2 and S < F : G is almost-simple ∙ S is non-abelian simple with Sylow p-subgroups of order p and ∙ G/S is soluble with Sylow p-subgroups of order p. The most difficult case to deal to rule out is when Soc(G) = F(G) is a cyclic group of order p and F∗(G)/F(G) a chief factor of G isomorphic to a non-abelian simple group with Sylow p-subgroups of order p. The p-soluble case. Theorem 3.10. ([5]) Let G be a p-soluble group. Then, G has property (†) if and only if one of the following holds. (1) G is p-supersoluble. (2) G is a group such that if P is a Sylow p-subgroup of G and Q is a 2-maximal subgroup of P , then Φ(G/Op′(G)) ≤ QOp′(G)/Op′(G); ∙ if Φ(G/Op′(G)) = QOp′(G)/Op′(G), then every chief series of the group G has exactly one complemented p-chief factor; moreover, this p-chief factor has order p2; ∙ if Φ(G/Op′(G)) < QOp′(G)/Op′(G), then all complemented p-chief factors of G are G-isomorphic to a 2-dimensional ir- reducible G-module V which is not an absolutely irreducible G-module. Why does it appear the absolutely irreducible module? In a minimal counterexample, we have Op′(G) = Φ(G) = 1 and F(G) is the Sylow p-subgroup of G. In other words, we have to analyse F(G). Now F(G) is a homogeneous G-module over GF(p) and if V is an irre- ducible submodule of F(G) and V is absolutely irreducible, then number Jo u rn al A lg eb ra D is cr et e M at h .26 On the cover-avoidance property of irreducible submodules, or, equivalently the number of minimal nor- mal subgroups of G is quite short and we do not have enough chief series to fulfill the condition by which every 2-maximal subgroup of F(G) is a partial CAP-subgroup of G. The Theorem. We bring the article to a close with the characterisation of groups enjoying property (†). Theorem 3.11. ([5]) Let G be a group and P ∈ Sylp(G). Assume that p2 divides ∣G∣. Then G has property (†) if and only if G satisfies one of the following conditions: (1) G is p-supersoluble; (2) G is p-soluble and if Q is 2-maximal in P , then Φ(G/Op′(G)) ≤ QOp′(G)/Op′(G); ∙ if Φ(G/Op′(G)) = QOp′(G)/Op′(G), then every chief series of the group G has exactly one complemented p-chief factor; moreover, this p-chief factor has order p2; ∙ if Φ(G/Op′(G)) < QOp′(G)/Op′(G), then all complemented p-chief factors of G are G-isomorphic to a 2-dimensional ir- reducible G-module V which is not an absolutely irreducible G-module. (3) G is non p-soluble and ∣P ∣ = p2; (4) p = 2 and G is non 2-soluble, and P is isomorphic to Q8. References [1] R. K. Agrawal, The influence on a finite group of its permutable subgroups. Canad. Math. Bull. 17 (1974) 159–165. [2] A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite groups, Springer, Dordrecht, 2006. [3] A. Ballester-Bolinches, Luis M. Ezquerro and Alexander N. Skiba, Local embed- dings of some families of subgroups of finite groups, Acta Math. Sinica, English Series 25(7) (2009) 869–882. [4] A. Ballester-Bolinches, Luis M. Ezquerro and Alexander N. Skiba, Subgroups of finite groups with a strong cover-avoidance property. Bull. of the Austral. Math. Soc. 79 (2009) 499-506. Jo u rn al A lg eb ra D is cr et e M at h .A. Ballester-Bolinches, L. M. Ezquerro, A. N. Skiba 27 [5] A. Ballester-Bolinches, Luis M. Ezquerro and Alexander N. Skiba, On second maximal subgroups of Sylow subgroups of finite groups. Submitted. [6] R. W. Carter and T. O. Hawkes, The F-normalizers of a finite soluble group J. Algebra 5 (1967) 175–201. [7] Y.Cui and and X. Guo. Influence of the Semi Cover-Avoiding property onf the Structure of Finite Groups . International J. Algebra 2 2008, 395-402. [8] K. Doerk and T. Hawkes, Finite Soluble Groups. Walter de Gruyter, Berlin-New York, 1992. [9] L. M. Ezquerro, A contribution to the theory of finite supersoluble groups, Rend. Sem. Mat. Univ. Padova 89 (1993) 161–170. [10] W. Gaschütz. Prefrattinigruppen. Arch. Math. 13, 1962, 418-426. [11] Y. Fan, X. Guo and K. P. Shum. Remarks on two generalizations of normality of subgroups. Chinese J. Contemp. Math. 27 2006, 139-146. [12] R. L. Griess and P. Schmid, The Frattini module. Arch. Math. 30 (1978) 256–266. [13] X. Guo and K. P. Shum, Cover-avoidance properties and the structure of finite groups. J. Pure and Applied Algebra 181 (2003) 297–308. [14] X. Guo, Junxin Wang and K. P. Shum. On Semi-Cover-Avoiding Maximal sub- groups and Solvability of Finite Groups. Comm. Algebra 34, 2006, 3235-3244. [15] X. Guo, P. Guo and K. P. Shum. On semi cover-avoiding subgroups of finite groups. J. Pure and Applied Algebra 209, 2007, 151-158. [16] W. Guo, K. P. Shum and A. N. Skiba, X-semipermutable subgroups of finite groups. J. Algebra 315 (2007), no. 1, 31–41. [17] X. Y. Guo and L. L. Wang. On finite groups with some semi cover-avoiding subgroups. Acta Math. Sin. (Engl. Ser.) 23, 2007, 1689-1696 [18] P. Hall, A note on soluble groups J. London Math. Soc. 3 (1928) 98–105. [19] P. Hall, On the system normalizers of a soluble group. Proc. London Math. Soc.(2) 43 (1937) 507–528. [20] B. Hartley. On Fischer’s dualization of formation theory. Proc. London Math. Soc.(3) 19 1969, 193–207. [21] B. Huppert, Normalteiler und maximale Untergruppen endlicher Gruppen. Math. Z. 60 (1954) 409–434. [22] B. Huppert, Endliche Gruppen I. Die Grundlehren der Mathematischen Wis- senschaften 134 Springer-Verlag, Berlin-New York 1967. [23] B. Huppert and N. Blackburn, Finite Groups III. Grundlehren der Mathematis- chen Wissenschaften 243. Springer-Verlag, Berlin-New York, 1982. [24] D. L. Johnson. A note on supersoluble groups. Can. J. Math. 23, 1971, 562-564. [25] A. Mann, Finite groups whose n-maximal subgroups are subnormal. Trans. Amer. Math. Soc. 132 (1968) 395–409. [26] Li Shirong, Finite non-nilpotent groups all of whose second maximal subgroups are TI-groups. Math. Proc. R. Ir. Acad. 100A (2000) 65–71. [27] Y. M. Wang. C-normality of groups and its properties. J. Algebra 180, 1996, 954-965. [28] L. Wang and G. Chen. Some properties of finite groups with some (semi-p-)cover- avoiding subgroups. J. Pure and Applied Algebra 213, 2009, 686–689. Jo u rn al A lg eb ra D is cr et e M at h .28 On the cover-avoidance property Contact information A. Ballester- Bolinches Departament d’Àlgebra, Universitat de València, Dr. Moliner, 50, E-46100 Burjas- sot, València, Spain E-Mail: Adolfo.Ballester@uv.es L. M. Ezquerro Departamento of Matemáticas, Universidad Pública de Navarra, Campus de Arrosad́ıa, E-31006 Pamplona, Navarra, Spain E-Mail: ezquerro@unavarra.es A. N. Skiba Department of Mathematics, Gomel State University F. Skorina, Gomel 246019, Be- larus E-Mail: alexander.skiba49@gmail.com Received by the editors: 25.07.2009 and in final form 25.07.2009.

On subgroups which cover or avoid chief factors of a finite group

Institution

Ähnliche Einträge