Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T11:55:54.659Z Has data issue: false hasContentIssue false
  • English
  • Français

Group algebras with an Engel group of units

Part of: Rings and algebras arising under various constructions Representation theory of groups Conditions on elements

Published online by Cambridge University Press:  09 April 2009

A. Bovdi
A. Bovdi
Affiliation:
University of Debrecen, 4010 Debrecen, Hungary, e-mail: [email protected]
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.

Let F be a field of characteristic p and G a group containing at least one element of order p. It is proved that the group of units of the group algebra FG is a bounded Engel group if and only if FG is a bounded Engel algebra, and that this is the case if and only if G is nilpotent and has a normal subgroup H such that both the factor group G/H and the commutator subgroup H′ are finite p–groups.

Keywords

Group algebragroup of unitsn–Engel groupsEngel groups.

MSC classification

Secondary: 16S34: Group rings , Laurent polynomial rings 16U60: Units, groups of units 20C05: Group rings of finite groups and their modules
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 2 , April 2006 , pp. 173 - 178
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Bovdi, A. A. and Khripta, I. I., ‘Engel properties of the multiplicative group of a group algebra’, Doklady Akad. Nauk SSSR 314 (1990), 18–20 (Russian);Google Scholar
English translation: Soviet Math. Dokl. 42 (1991), 243–246.Google Scholar
[2]Bovdi, A. A. and Khripta, I. I., ‘Engel properties of the multiplicative group of a group algebra’, Mat. Sb. 182 (1991), 130–144Google Scholar
(Russian); English translation: Math. USSR Sbornik 72 (1992), 121–134.CrossRefGoogle Scholar
[3]Fischer, J. L., Parmenter, M. M. and Sehgal, S. K., ‘Group rings with solvable n–Engel unit groups’, Proc. Amer. Math. Soc. 59 (1976), 195–200.Google Scholar
[4]Giambruno, A., Sehgal, S. K. and Valenti, A., ‘Group identities on units of algebras’, J. Algebra 226 (2000), 488–504.CrossRefGoogle Scholar
[5]Khripta, I. I., ‘The nilpotency of the multiplicative group of a group ring’, Mat. Zametki 11 (1972), 191–200 (Russian);Google Scholar
English translation: Math. Notes 11 (1972), 119–124.CrossRefGoogle Scholar
[6]Liu, C.-H. and Passman, D. S., ‘Group algebras with units satisfying a group identity, II’, Proc. Amer. Math. Soc. 127 (1999), 337–341.CrossRefGoogle Scholar
[7]Plotkin, B. I., Groups of automorphism of algebraic systems (Nauka, Moscow, 1966) (in Russian); English translation: (Wolters-Nordhoff, Groningen, 1972).Google Scholar
[8]Robinson, D. J. S., A course in the theory of groups, 2nd edition (Springer, New York, 1996).CrossRefGoogle Scholar
[9]Sehgal, S. K., Topics in group rigs (Marcel Dekker, New York, 1978).Google Scholar
[10]Shalev, A., ‘On associative algebras satisfying the Engel condition’, Israel J. Math. 67 (1989), 287–289.CrossRefGoogle Scholar
[11]Strojnowksi, A., ‘A note on u.p. groups’, Comm. Algebra 8 (1980), 231–234.CrossRefGoogle Scholar