# On finite minimal non-p-supersoluble groups

Colloquium Mathematicae (1992)

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

## Access Full Article

top## Abstract

top## How to cite

topTuccillo, 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] R. Carter, B. Fisher and T. Hawkes, Extreme classes of finite soluble groups, J. Algebra 9 (1968), 285-313. Zbl0177.03902
- [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] E. L. Dickson, Linear Groups with an Exposition of the Galois Field Theory, Teubner, Leipzig 1901 (Dover reprint 1958). Zbl32.0128.01
- [4] K. Doerk, Minimal nicht überauflösbare, endliche Gruppen, Math. Z. 91 (1966), 198-205. Zbl0135.05401
- [5] W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029. Zbl0124.26402
- [6] D. Gorenstein, Finite Groups, Harper and Row, New York 1968.
- [7] B. Huppert, Endliche Gruppen I, Springer, Berlin 1967. Zbl0217.07201
- [8] B. Huppert and N. Blackburn, Finite Groups III, Springer, Berlin 1982. Zbl0514.20002
- [9] N. Ito, Note on (LM)-groups of finite orders, Kōdai Math. Sem. Reports 1951, 1-6. Zbl0044.01303
- [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. Zbl0075.24003
- [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. Zbl0463.20018
- [14] M. Suzuki, On a class of doubly transitive groups, Ann. of Math. 75 (1962), 105-145. Zbl0106.24702
- [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 ?

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