Abstract
Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is called an \(\mathcal{H }\)-subgroup of \(G\) if \(N_G(H)\cap H^g\le H\) for all \(g\in G\). A group \(G\) is said to be an \({\mathcal{H }}_p\)-group if every cyclic subgroup of \(G\) of prime order or order 4 is an \(\mathcal{H }\)-subgroup of \(G\). In this paper, the structure of a finite group all of whose second maximal subgroups are \({\mathcal{H }}_p\)-subgroups has been characterized.
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References
Bianchi, M., Gillio Berta Mauri, A., Herzog, M., Verardi, L.: On finite sovlable groups in which normality is a transitive relation. J. Group Theory 3(2), 147–156 (2000)
Chen, Z.: Inner supersolvable groups of finite order. J. Southwest Teachers College 1, 2–14 (1983)
Csörgö, P., Herzog, M.: On supersolvable groups and the nilpotator. Comm. Algebra 32(2), 609–620 (2004)
Gorenstein, D.: Finite Groups. Harper & Row, New York (1968)
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)
Li, S.R.: On minimal non-\(PE\)-groups. J. Pure Appl. Algebra 132(2), 149–158 (1998)
Li, S.R., Zhao, Y.Q.: Some finite non-solvable groups characterized by their solvable subgroups. Acta Math. Sin. (New Ser) 4(8), 5–13 (1988)
Li, Y.M.: Finite groups with \(NE\)-subgroups. J. Group Theory 9, 49–58 (2006)
Lu, J.K., Guo, X.Y.: Notes on \(NE\)-subgroups of finite groups. Front. Math. China 5(4), 679–685 (2010)
Robinson, D.J.S.: A Course in the Theory of Groups. Springer, New York (1996)
Sastry, N.: On minimal non-\(PN\)-groups. J. Algebra 65, 104–109 (1980)
Zhong, X.G., Li, S.R.: Finite groups whose second maximal subgroups are \({\cal H}\)-QC-subgroups. Comm. Algebra 37(6), 2121–2126 (2009)
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The research of the authors is supported by the National Natural Science Foundation of China(11261007), the Science Foundation of Guangxi Autonomous Region(0991090) and Guangxi Special Funds for Discipline Construction of Degree Programs.
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Communicated by J. S. Wilson.
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Zhong, X., Lu, J. & Li, Y. Finite groups all of whose second maximal subgroups are \({\mathcal{H }}_p\)-groups. Monatsh Math 172, 477–486 (2013). https://doi.org/10.1007/s00605-013-0478-1
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DOI: https://doi.org/10.1007/s00605-013-0478-1