On minimal non-[inline-graphic not available: see fulltext]-groups

Abstract

A finite group G is called an [inline-graphic not available: see fulltext]-group if every proper subgroup H of G is either subnormal in G or self-normalizing. We determinate the structure of non-[inline-graphic not available: see fulltext]-groups in which all proper subgroups are [inline-graphic not available: see fulltext] groups.

Keywords

Subnormal subgroup / self-normalizing subgroup / [inline-graphic not available: see fulltext]-group / minimal non-[inline-graphic not available: see fulltext]-group

References

[1] Doerk K. Minimal nicht überauflösbare endliche Gruppen. Math Z, 1966, 91: 198-205

[2] Guo P, Guo X. On minimal non-MSN-groups. Front Math China, 2011, 6(5): 847-854

[3] Itô N. Über die Frattini-Gruppe einer endlichen Gruppe. Proc Japan Acad, 1955, 31: 327-328

[4] King O H. The subgroup structure of finite classical groups in terms of geometric configurations. Surveys in Combinatorics, 2005, Cambridge: Cambridge University Press 29 56

[5] Kurzweil H, Stellmacher B. The Theory of Finite Groups (An Introduction), 2004, New York: Springer

[6] Laffey T J. A lemma on finite p-group and some consequences. Proc Cambr Phil Soc, 1974, 75: 133-137

[7] Liu X, Xiao G. Finite groups with only subnormal or self-normal subgroups. J Sichuan Normal Univ, 2000, 23(5): 479-481

[8] Miller G A, Moreno H C. Non-abelian groups in which every subgroup is abelian. Trans Amer Math Soc, 1903, 4: 398-404

[9] Sastry N. On minimal non-PN-groups. J Algebra, 1980, 65: 104-109

[10] Suzuki M. On a class of doubly transitive groups. Ann Math, 1962, 75(1): 105-145

[11] Thompson J G. Nonsolvable finite groups all of whose local subgroups are solvable I. Bull Amer Math Soc, 1968, 74: 383-437