具有小秩Sylow-子群的有限可解群
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摘要
设G是有限p-群,■,其中E为G的初等交换子群,称为G的秩。再令■,其中E为G的初等交换子群,称为G的正规秩。研究Sylow-子群的正规秩≤3的可解群的结构问题,运用极小阶反例法,证明了若G为有限可解群且G的Sylow-子群的正规秩≤3,则G∈N_(2′)N_(2′)N_2U。更进一步,群G的幂零长不超过5,且对所有的素数p,G的p-长不超过2。
We assume G is a finite p-group, r(G)=max{log_p|E|,E≤G}.Wherein, E is an elementary exchange subgroup of G, called rank of G. We make r_n(G)=max{log_p|E|,E≤G,E?G}, where E is an elementary called normal rank of G. The structure problem of solvable groups of normal rank of Sylow-subgroups ≤3 was studied. The method of minimal counterexample was used to prove If G is a finite soluble group and the normal rank of Sylow-subgroup of G≤3, then G∈N_(2′)N_(2′)N_2U. Moreover, the nilpotent length of G is no more than 5 and for every prime numbe p, the p-length of G is no more than 2.
We assume G is a finite p-group, r(G)=max{log_p|E|,E≤G}.Wherein, E is an elementary exchange subgroup of G, called rank of G. We make r_n(G)=max{log_p|E|,E≤G,E?G}, where E is an elementary called normal rank of G. The structure problem of solvable groups of normal rank of Sylow-subgroups ≤3 was studied. The method of minimal counterexample was used to prove If G is a finite soluble group and the normal rank of Sylow-subgroup of G≤3, then G∈N_(2′)N_(2′)N_2U. Moreover, the nilpotent length of G is no more than 5 and for every prime numbe p, the p-length of G is no more than 2.
引文
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[3] Janko Z. A classification of finite 2-groups with exactly three involutions [J]. Algebra, 2006, 291(2), 505-533.
[4] Chillag D, Sonn J. Sylow-metacyclic groups and Q-admissibility [J]. Israel J. Math,1981, 40 (3/4): 307-323.
[5] Huppert B. Endliche Gruppen:I [M]. Berlin-Heidelberg: Springer-Verlag,1967.
[6] Gaschutz W. Lectures on subgroups of Sylow type in finite soluble groups [M]. Canberra: Australian National University, 1979.
[7] Shemetkov L A. Formations of Finite Groups [M]. Moscow:Nauka, 1978.
[8] Mazurov V D. Finite groups with metacyclic Sylow 2-subgroups [J]. Siberian Mathematical Journal, 1967, 8(4):733-733.
[9] Janko Z. Elements of order at most 4 in finite 2-groups [J]. Journal of Group Theory, 2004, 7(4):431-436.
[2] Blackburn N. Generalizations of certain elementary theoerms on p-groups [J]. London Math, 1961, 11(3):1-22.
[3] Janko Z. A classification of finite 2-groups with exactly three involutions [J]. Algebra, 2006, 291(2), 505-533.
[4] Chillag D, Sonn J. Sylow-metacyclic groups and Q-admissibility [J]. Israel J. Math,1981, 40 (3/4): 307-323.
[5] Huppert B. Endliche Gruppen:I [M]. Berlin-Heidelberg: Springer-Verlag,1967.
[6] Gaschutz W. Lectures on subgroups of Sylow type in finite soluble groups [M]. Canberra: Australian National University, 1979.
[7] Shemetkov L A. Formations of Finite Groups [M]. Moscow:Nauka, 1978.
[8] Mazurov V D. Finite groups with metacyclic Sylow 2-subgroups [J]. Siberian Mathematical Journal, 1967, 8(4):733-733.
[9] Janko Z. Elements of order at most 4 in finite 2-groups [J]. Journal of Group Theory, 2004, 7(4):431-436.