FINITE GROUPS WHOSE MINIMAL SUBGROUPS ARE WEAKLY H-SUBGROUPS

doi:10.1016/S0252-9602(12)60179-9

Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (6): 2295-2301.doi: 10.1016/S0252-9602(12)60179-9

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FINITE GROUPS WHOSE MINIMAL SUBGROUPS ARE WEAKLY H-SUBGROUPS

M. M. Al-Mosa Al-Shomrani|M. Ramadan|A. A. Heliel

1.Department of Mathematics, Faculty of Science 80203, King Abdulaziz University, Jeddah 21589, Saudi Arabia； 2.Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt； 3.Department of Mathematics, Faculty of Science 80203, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics, Beni-Suef University, Faculty of Science 62511, Beni-Suef, Egypt

Received:2011-06-09 Revised:2012-03-21 Online:2012-11-20 Published:2012-11-20
Supported by:
The research supported by the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) represented by the Unit of Research Groups through the grant number (MG/31/01) for the group entitled “Abstract Algebra and its Applications”.

Abstract

Abstract:

Let G be a finite group. A subgroup H of G is called an H-subgroup in G if N_G(H)∩H^g ≤ H for all g ∈ G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H∩K is an H-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G. Our results improve and generalize several recent results in the literature.

Key words: c-normal subgroup, H-subgroup, p-nilpotent group, supersolvable group, generalized Fitting subgroup, saturated formation

CLC Number:

20D10

M. M. Al-Mosa Al-Shomrani, M. Ramadan, A. A. Heliel. FINITE GROUPS WHOSE MINIMAL SUBGROUPS ARE WEAKLY H-SUBGROUPS[J].Acta mathematica scientia,Series B, 2012, 32(6): 2295-2301.

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References

[1] Asaad M. On p-nilpotence and supersolvability of finite groups. Comm Algebra, 2006, 34: 189–195

[2] Asaad M, Ezzat Mohamed M. On c-normality of finite groups. J Aust Math Soc, 2005, 78: 297–304

[3] Asaad M, Heliel A A, Al-Mosa Al-Shomrani M M. On weakly H-subgroups of finite groups. Comm Algebra, 2012, DOI: 0.1080/00927872.2011.591218

[4] Asaad M, Ramadan M. Finite groups whose minimal subgroups are c-supplemented. Comm Algebra, 2008, 36: 1034–1040

[5] Asaad M, Li S. On minimal subgroups of finite groups II. Comm Algebra, 1996, 24(14): 4603–4606

[6] Buckley J. Finite groups whose minimal subgroups are normal. Math Z, 1970, 116: 15–17

[7] Ballester-Bolinches A, Wang Y. Finite groups with some c-normal minimal subgroups. J Pure Appl Algebra, 2000, 153: 121–127

[8] Bianchi M, Gillio Berta Mauri A, Herzog M, Verardi L. On finite solvable groups in which normality is a transitive relation. J Group Theory, 2000, 3: 147–156

[9] Cs¨org¨o P, Herzog M. On supersolvable groups and the nilpotator. Comm Algebra, 2004, 32: 609–620

[10] Doerk K, Hawkes T. Finite Soluble Groups. Berlin, New York: Walter de Gruyter, 1992

[11] Guo X, Wei X. The influence of H-subgroups on the structure of finite groups. J Group Theory, 2010, 13: 267–276

[12] Huppert B. Endliche Gruppen I. Berlin, Heidelberg, New York: Springer-Verlage, 1979

[13] Huppert B, Blackburn N. Finite Groups III. Berlin, Heidelberg, New York: Springer-Verlage, 1982

[14] Li D, Guo X. The influence of c-normality of subgroups on the structure of finite groups II. Comm Algebra, 1998, 26(6): 1913–1922

[15] Li D, Guo X. The influence of c-normality of subgroups on the structure of finite groups. J Pure Appl Algebra, 2000, 150: 53–60

[16] Ramadan M, Ezzat Mohamed M, Heliel A A. On c-normality of certain subgroups of prime power order of finite groups. Arch Math, 2005, 85: 203–210

[17] Wang Y. C-normality of groups and its properties. J Algebra, 1996, 180: 954–965

[18] Wang Y. The influence of minimal subgroups on the structure of finite groups. Acta Math Sin, 2000, 16(1): 63–70

[19] Wei H Q. On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups. Comm Algebra, 2001, 29(5): 2193–2200

[20] Wei H Q, Wang Y M, Li Y M. On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups II. Comm Algebra, 2003, 31(10): 4807–4816

[21] Weinstein M (editor). Between Nilpotent and Solvable. Passaic: Polygonal Publishing House, 1982

[22] Li Y M. Some notes on the minimal subgroups of Fitting subgroups of finite groups. J Pure Appl Algebra, 2002, 171: 289–294

FINITE GROUPS WHOSE MINIMAL SUBGROUPS ARE WEAKLY H-SUBGROUPS

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