Geodesic
ON WEAKLY -SUPPLEMENTED SUBGROUPS OF SYLOW p-SUBGROUPS OF FINITE GROUPS*
Glasgow mathematical journal, Tome 53 (2011) no. 2, pp. 401-410

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DOI

A subgroup H is called weakly -supplemented in a finite group G if there exists a subgroup B of G provided that (1) G = HB, and (2) if H1/HG is a maximal subgroup of H/HG, then H1B = BH1 < G, where HG is the largest normal subgroup of G contained in H. In this paper we will prove the following: Let G be a finite group and P be a Sylow p-subgroup of G, where p is the smallest prime divisor of |G|. Suppose that P has a non-trivial proper subgroup D such that all subgroups E of P with order |D| and 2|D| (if P is a non-abelian 2-group, |P : D| > 2 and there exists D1 ⊴ E ≤ P with 2|D1| = |D| and E/D1 is cyclic of order 4) have p-nilpotent supplement or weak -supplement in G, then G is p-nilpotent.
DOI : 10.1017/S0017089511000036
Mots-clés : 20D10, 20D20 
MIAO, LONG. ON WEAKLY -SUPPLEMENTED SUBGROUPS OF SYLOW p-SUBGROUPS OF FINITE GROUPS*. Glasgow mathematical journal, Tome 53 (2011) no. 2, pp. 401-410. doi: 10.1017/S0017089511000036
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     author = {MIAO, LONG},
     title = {ON {WEAKLY} {-SUPPLEMENTED} {SUBGROUPS} {OF} {SYLOW} {p-SUBGROUPS} {OF} {FINITE} {GROUPS*}},
     journal = {Glasgow mathematical journal},
     pages = {401--410},
     year = {2011},
     volume = {53},
     number = {2},
     doi = {10.1017/S0017089511000036},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089511000036/}
}
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