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1992 | 63 | 1 | 119-131
Tytuł artykułu

On finite minimal non-p-supersoluble groups

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
If ℱ is a class of groups, then a minimal non-ℱ-group (a dual minimal non-ℱ-group resp.) is a group which is not in ℱ but any of its proper subgroups (factor groups resp.) is in ℱ. In many problems of classification of groups it is sometimes useful to know structure properties of classes of minimal non-ℱ-groups and dual minimal non-ℱ-groups. In fact, the literature on group theory contains many results directed to classify some of the most remarkable among the aforesaid classes. In particular, V. N. Semenchuk in [12] and [13] examined the structure of minimal non-ℱ-groups for ℱ a formation, proving, among other results, that if ℱ is a saturated formation, then the structure of finite soluble, minimal non-ℱ-groups can be determined provided that the structure of finite soluble, minimal non-ℱ-groups with trivial Frattini subgroup is known. In this paper we use this result with regard to the formation of p-supersoluble groups (p prime), starting from the classification of finite soluble, minimal non-p-supersoluble groups with trivial Frattini subgroup given by N. P. Kontorovich and V. P. Nagrebetskiĭ ([10]). The second part of this paper deals with non-soluble, minimal non-p-supersoluble finite groups. The problem is reduced to the case of simple groups. We classify the simple, minimal non-p-supersoluble groups, p being the smallest odd prime divisor of the group order, and provide a characterization of minimal simple groups. All the groups considered are finite.
Słowa kluczowe
Rocznik
Tom
63
Numer
1
Strony
119-131
Opis fizyczny
Daty
wydano
1992
otrzymano
1990-01-03
poprawiono
1991-04-08
Twórcy
  • Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università di Napoli, via Mezzocannone, 8, I-80134 Napoli, Italy
Bibliografia
  • [1] R. Carter, B. Fisher and T. Hawkes, Extreme classes of finite soluble groups, J. Algebra 9 (1968), 285-313.
  • [2] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford 1985.
  • [3] E. L. Dickson, Linear Groups with an Exposition of the Galois Field Theory, Teubner, Leipzig 1901 (Dover reprint 1958).
  • [4] K. Doerk, Minimal nicht überauflösbare, endliche Gruppen, Math. Z. 91 (1966), 198-205.
  • [5] W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029.
  • [6] D. Gorenstein, Finite Groups, Harper and Row, New York 1968.
  • [7] B. Huppert, Endliche Gruppen I, Springer, Berlin 1967.
  • [8] B. Huppert and N. Blackburn, Finite Groups III, Springer, Berlin 1982.
  • [9] N. Ito, Note on (LM)-groups of finite orders, Kōdai Math. Sem. Reports 1951, 1-6.
  • [10] N. P. Kontorovich and V. P. Nagrebetskiĭ, Finite minimal non-p-supersolvable groups, Ural. Gos. Univ. Mat. Zap. 9 (1975), 53-59, 134-135 (in Russian).
  • [11] L. Rédei, Die endlichen einstufig nichtnilpotenten Gruppen, Publ. Math. Debrecen 4 (1956), 303-324.
  • [12] V. N. Semenchuk, Minimal non-ℱ-groups, Dokl. Akad. Nauk BSSR 22 (7) (1978), 596-599 (in Russian).
  • [13] V. N. Semenchuk, Minimal non-ℱ-groups, Algebra i Logika 18 (3) (1979), 348-382 (in Russian); English transl.: Algebra and Logic 18 (3) (1979), 214-233.
  • [14] M. Suzuki, On a class of doubly transitive groups, Ann. of Math. 75 (1962), 105-145.
  • [15] J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383-437.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-cmv63i1p119bwm
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