On finite minimal non-p-supersoluble groups

Fernando Tuccillo

Colloquium Mathematicae (1992)

  • Volume: 63, Issue: 1, page 119-131
  • ISSN: 0010-1354

Abstract

top
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.

How to cite

top

Tuccillo, Fernando. "On finite minimal non-p-supersoluble groups." Colloquium Mathematicae 63.1 (1992): 119-131. <http://eudml.org/doc/210126>.

@article{Tuccillo1992,
abstract = {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.},
author = {Tuccillo, Fernando},
journal = {Colloquium Mathematicae},
keywords = {finite -supersolvable groups; minimal non--groups; saturated formation; Frattini subgroups; simple groups; minimal non-- supersolvable groups},
language = {eng},
number = {1},
pages = {119-131},
title = {On finite minimal non-p-supersoluble groups},
url = {http://eudml.org/doc/210126},
volume = {63},
year = {1992},
}

TY - JOUR
AU - Tuccillo, Fernando
TI - On finite minimal non-p-supersoluble groups
JO - Colloquium Mathematicae
PY - 1992
VL - 63
IS - 1
SP - 119
EP - 131
AB - 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.
LA - eng
KW - finite -supersolvable groups; minimal non--groups; saturated formation; Frattini subgroups; simple groups; minimal non-- supersolvable groups
UR - http://eudml.org/doc/210126
ER -

References

top
  1. [1] R. Carter, B. Fisher and T. Hawkes, Extreme classes of finite soluble groups, J. Algebra 9 (1968), 285-313. Zbl0177.03902
  2. [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. Zbl0568.20001
  3. [3] E. L. Dickson, Linear Groups with an Exposition of the Galois Field Theory, Teubner, Leipzig 1901 (Dover reprint 1958). Zbl32.0128.01
  4. [4] K. Doerk, Minimal nicht überauflösbare, endliche Gruppen, Math. Z. 91 (1966), 198-205. Zbl0135.05401
  5. [5] W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029. Zbl0124.26402
  6. [6] D. Gorenstein, Finite Groups, Harper and Row, New York 1968. 
  7. [7] B. Huppert, Endliche Gruppen I, Springer, Berlin 1967. Zbl0217.07201
  8. [8] B. Huppert and N. Blackburn, Finite Groups III, Springer, Berlin 1982. Zbl0514.20002
  9. [9] N. Ito, Note on (LM)-groups of finite orders, Kōdai Math. Sem. Reports 1951, 1-6. Zbl0044.01303
  10. [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. [11] L. Rédei, Die endlichen einstufig nichtnilpotenten Gruppen, Publ. Math. Debrecen 4 (1956), 303-324. Zbl0075.24003
  12. [12] V. N. Semenchuk, Minimal non-ℱ-groups, Dokl. Akad. Nauk BSSR 22 (7) (1978), 596-599 (in Russian). 
  13. [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. Zbl0463.20018
  14. [14] M. Suzuki, On a class of doubly transitive groups, Ann. of Math. 75 (1962), 105-145. Zbl0106.24702
  15. [15] J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383-437. Zbl0159.30804

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.