Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-07-07T03:03:16.838Z Has data issue: false hasContentIssue false
  • English
  • Français

On groups with small Engel depth

Published online by Cambridge University Press:  17 April 2009

Rolf Brandl
Rolf Brandl
Affiliation:
Mathematisches Institut, Am Hubland 12, D–8700 Würzburg, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Every finite group G satisfies a law [x, ry] = [x, sy] for some positive integers r < s. The minimal value of r is called the depth of G. It is well known that groups of depth 1 are abelian. In this paper we prove the following. Let G be a finite group of depth 2. Then G/F(G) is supersoluble, metabelian and has abelian Sylow p-subgroups for all odd primes p. Moreover, lp(G) ≤ 1 for p odd and l2(G2) ≤ 1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 28 , Issue 1 , August 1983 , pp. 101 - 110
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Brandl, Rolf, “A characterization of finite p-soluble groups of p-length one by commutator identities”, J. Austral. Math. Soc. Ser. A 30 (1980–1981), 257–263.CrossRefGoogle Scholar
[2]Brandl, Rolf, “Zur Theorie der untergruppenabgeschlossenen Formationen: Endlich Varietäten”, J. Algebra 73 (1981), 1–22.CrossRefGoogle Scholar
[3]Brandl, Rolf, “On finite abelian-by-nilpotent groups”, J. Algebra (to appear).Google Scholar
[4]Brandl, Rolf, “Infinite soluble groups with Engel cycles; a finiteness condition”, Math. Z. 182 (1983), 259–264.CrossRefGoogle Scholar
[5]Doerk, Klaus, “Minimal nicht überauflösbare, endliche Gruppen”, Math. Z. 91 (1966), 198–205.CrossRefGoogle Scholar
[6]Gorenstein, Daniel, Finite groups (Harper and Row, New York, Evanston, London, 1968).Google Scholar
[7]Gupta, N.D., “Some group laws equivalent to the commutative law”, Arch. Math. (Basel) 17 (1966), 97–102.CrossRefGoogle Scholar
[8]Huppert, B., Endliche Gruppen I (Die Grundlehren der Mathematischen Wissenschaften, 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[9]Николова, Д. [D. Nikolova], “Тождества в метабеленых многобразий AkA1” [Identities in the metabelian variety AkA 1], Serdica (to appear).Google Scholar
[10]РазмЫслов, Ю.П. [Ju.P. Razmyslov], “О проблема Холл–Хигмен” [He Hall–Higman problem], Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 833–847.Google ScholarPubMed
[11]Thompson, John G., “Nonsolvable finite groups all of whose local subgroups are solvable”, Bull. Amer. Math. Soc. 74 (1968), 383–437.CrossRefGoogle Scholar
[12]Wright, C.R.B., “On the nilpotency class of a group of exponent four”, Pacific J. Math. 11 (1961), 387–394.CrossRefGoogle Scholar