Cover-avoidance properties and the structure of finite groups. (English) Zbl 1028.20014

Gua, Xiuyun; Shum, K. P.

J. Pure Appl. Algebra 181, No. 2-3, 297-308 (2003).

Let \(G\) be a finite group. The authors call a subgroup \(A\) of the group \(G\) a CAP-subgroup of \(G\) if for any chief factor \(H/K\) of \(G\) \(H\cap A=K\cap A\) or \(HA=KA\). Some characterizations for a finite group to be solvable are obtained when some of its maximal subgroups or \(2\)-maximal subgroups are CAP-subgroups. In particular, it is proved that a group \(G\) is solvable if and only if there exists a solvable \(2\)-maximal subgroup \(L\) of \(G\) such that \(L\) is a CAP-subgroup of \(G\).
It is also proved that \(G\) is \(p\)-solvable if and only if a Sylow \(p\)-subgroup \(P\) of \(G\) is a CAP-subgroup of \(G\).

Reviewer: N.M.Kurnosenko (Gomel)

MSC:

20D10	Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20	Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D30	Series and lattices of subgroups
20E28	Maximal subgroups

Keywords:

\(p\)-solvability; \(p\)-nilpotency; finite groups; maximal subgroups; CAP-subgroups; Sylow subgroups

References:

[1]	Ezquerro, L.M., A contribution to the theory of finite supersolvable groups, Rend. sem. mat. univ. Padova, 89, 161-170, (1993) · Zbl 0809.20009
[2]	Gaschütz, W., Praefrattini gruppen, Arch. math., 13, 418-426, (1962)
[3]	Gillam, J.D., Cover-avoid subgroups in finite solvable groups, J. algebra, 29, 324-329, (1974) · Zbl 0293.20017
[4]	Gorenstein, D., Finite groups, (1968), Harper and Row Publishers New York, Evanston, London · Zbl 0185.05701
[5]	B. Huppert, Endliche Gruppen, Vol. I, Springer, New York, Berlin, 1967.
[6]	Kurzweil, H., Endliche gruppen, (1967), Springer New York
[7]	Robinson, D.J.S., A course in the theory of groups, (1993), Springer New York, Berlin · Zbl 0496.20038
[8]	Rose, J., On finite insolvable groups with nilpotent maximal subgroups, J. algebra, 48, 182-196, (1977) · Zbl 0364.20034
[9]	K.U. Schaller, Über Deck-Meide-Untergruppen in endlichen auflösbaren Gruppen, Ph.D. Thesis, University of Kiel, 1971.
[10]	Tomkinson, M.J., Cover-avoidance properties in finite soluble groups, Canad. math. bull., 19, 2, 213-216, (1976) · Zbl 0352.20018