×

The structure of finite non-nilpotent groups in which every 2-maximal subgroup permutes with all 3-maximal subgroups. (English) Zbl 1194.20021

Subgroups \(H\) of a group \(G\) which are maximal subgroups of maximal subgroups are called \(2\)-maximal or second-maximal subgroups. Analogously we can define \(3\)-maximal subgroups, \(4\)-maximal subgroups, and so on. A natural question in this setting is the influence of \(i\)-maximal subgroups on the structure of the groups.
The authors describe first the groups in which every \(2\)-maximal subgroup permutes with all maximal subgroups. These groups \(G\) are either nilpotent, or are supersoluble of order \(pq^\beta\) such that a Sylow \(q\)-subgroup \(Q\) of \(G\) is cyclic and its core \(Q_G=\langle x^q\rangle\) (Theorem 2.4). Then they describe the non-nilpotent groups in which every \(2\)-maximal subgroup permutes with all \(3\)-maximal subgroups. These are the groups whose order is the product of at most \(3\) primes (two of them perhaps the same) or \(G\) is a Shmidt group with Abelian Sylow subgroups or isomorphic to \(\text{SL}(2,3)\), or a supersoluble group of one of four types (Theorem 3.5). Finally, some examples are given to show that there are supersoluble groups of each of these four types.

MSC:

20D40 Products of subgroups of abstract finite groups
20E28 Maximal subgroups
20D25 Special subgroups (Frattini, Fitting, etc.)
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D15 Finite nilpotent groups, \(p\)-groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agrawal R. K., Proc. Amer. Math. Soc. 54 pp 13– (1976) · doi:10.1090/S0002-9939-1976-0409651-8
[2] Gagen T. M., J. Austral. Math. Soc. 6 pp 466– (1966) · Zbl 0166.02102 · doi:10.1017/S1446788700004936
[3] Gorenstein D., Finite Groups, 2. ed. (1980) · Zbl 0463.20012
[4] Guo W., The Theory of Classes of Groups (2000)
[5] Guo W., Southeast Asian Bulletin of Math. 29 pp 493– (2005)
[6] Guo W., Siberian Math. J. 48 pp 593– (2007) · doi:10.1007/s11202-007-0061-x
[7] Guo W., J. Algebra 315 pp 31– (2007) · Zbl 1130.20017 · doi:10.1016/j.jalgebra.2007.06.002
[8] Guo W., Siberian Math. J. 45 pp 433– (2004) · doi:10.1023/B:SIMJ.0000028608.59920.af
[9] Guo X., J. Pure Applied Algebra 181 pp 297– (2003) · Zbl 1028.20014 · doi:10.1016/S0022-4049(02)00327-4
[10] Huppert B., Math. Z. 60 pp 409– (1954) · Zbl 0057.25303 · doi:10.1007/BF01187387
[11] Huppert B., Endliche Gruppen I (1967) · Zbl 0217.07201
[12] Janko Z., Math. Z. 82 pp 82– (1963) · Zbl 0118.26704 · doi:10.1007/BF01112825
[13] Poljakov L. Ja, Finite Groups. (1966)
[14] Robinson D. J. S., A Course in the Theory of Groups (1982) · Zbl 0483.20001
[15] Shemetkov L. A., Formations of Finite Groups (1978) · Zbl 0496.20014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.