Abstract
Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever \({K\trianglelefteq M}\), then K G ∩ M = K where K G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every maximal subgroup of G is an NR-subgroup then G is solvable. This gives a positive answer to a conjecture posed in Berkovich (Houston J. Math. 24 (1998), 631–638).
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References
J. H. Conway et al., Atlas of Finite groups, Maximal subgroups and ordinary characters for simple groups, with computational assitance from J. G. Thackray, Oxford University Press, Eynsham, 1985.
Berkovich Y.: Subgroups with the Character Restriction Property and Related Topics. Houston J. Math. 24, 631–638 (1998)
R. W. Carter, Simple Groups of Lie Type, Pure and Applied Mathematics, 28. John Wiley and Sons, London-New York-Sydney, 1972.
D. Gorenstein, Finite Groups, Chelsea Publishing Company, 2nd ed., 1980.
Liebeck M.W., Praeger C.E., Saxl J.: A Classification of the Maximal Subgroups of the Finite Alternating and Symmetric Groups. J. Algebra 111, 365–383 (1987)
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This work was completed with the support of University of Birmingham.
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Tong-Viet, H.P. Groups with normal restriction property. Arch. Math. 93, 199–203 (2009). https://doi.org/10.1007/s00013-009-0030-7
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DOI: https://doi.org/10.1007/s00013-009-0030-7