ON A THEOREM OF HUPPERT
2011 ◽
Vol 10
(02)
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pp. 295-301
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A well-known theorem of Huppert states that a finite group is soluble if its every proper subgroup is supersoluble. In this paper, we proved the following result: let G be a finite group. (1) If G has exactly n non-supersoluble proper subgroups, where 0 ≤ n ≤ 7 and n ≠ 5, then G is soluble. (2) G is a non-soluble group with exactly five non-supersoluble proper subgroups if and only if all non-supersoluble proper subgroups are conjugate maximal subgroups and G/Φ(G) ≅ A5, where Φ(G) is the Frattini subgroup of G. Furthermore, we also considered the influence of the number of non-abelian proper subgroups on the solubility of finite groups.
1969 ◽
Vol 21
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pp. 418-429
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1975 ◽
Vol 77
(2)
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pp. 247-257
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1989 ◽
Vol 12
(2)
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pp. 263-266
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1997 ◽
Vol 40
(2)
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pp. 243-246
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2011 ◽
Vol 53
(2)
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pp. 401-410
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