Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T21:16:52.595Z Has data issue: false hasContentIssue false
  • English
  • Français

AXIOMATISABILITY OF THE CLASS OF MONOLITHIC GROUPS IN A VARIETY OF NILPOTENT GROUPS

Part of: Varieties Special aspects of infinite or finite groups Structure and classification of infinite or finite groups Representation theory of groups Foundations

Published online by Cambridge University Press:  16 January 2020

JOSHUA T. GRICE Open the ORCID record for JOSHUA T. GRICE [Opens in a new window]
JOSHUA T. GRICE*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC29208, USA email jtgrice@math.sc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The class of all monolithic (that is, subdirectly irreducible) groups belonging to a variety generated by a finite nilpotent group can be axiomatised by a finite set of elementary sentences.

Keywords

subdirectly irreducible groupmonolithic groupnilpotent groupvariety of groupsfinite axiomatisability

MSC classification

Primary: 20A05: Axiomatics and elementary properties
Secondary: 08B10: Congruence modularity, congruence distributivity 20D15: Nilpotent groups, $p$-groups 20E10: Quasivarieties and varieties of groups 20F18: Nilpotent groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 1 , August 2020 , pp. 67 - 76
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baker, K. A. and Wang, J., ‘Definable principal subcongruences’, Algebra Universalis 47(2) (2002), 145–151.CrossRefGoogle Scholar
Birkhoff, G., ‘On the structure of abstract algebras’, Math. Proc. Cambridge Philos. Soc. 31(4) (1935), 433–454.CrossRefGoogle Scholar
Birkhoff, G., ‘Subdirect unions in universal algebra’, Bull. Amer. Math. Soc. 50 (1944), 764–768.CrossRefGoogle Scholar
Dummit, D. S. and Foote, R. M., Abstract Algebra, 3rd edn (John Wiley & Sons, Hoboken, NJ, 2004).Google Scholar
Freese, R. and McKenzie, R., Commutator Theory for Congruence Modular Varieties, London Mathematical Society Lecture Note Series, 125 (Cambridge University Press, Cambridge, 1987).Google Scholar
Lyndon, R. C., ‘Two notes on nilpotent groups’, Proc. Amer. Math. Soc. 3 (1952), 579–579.CrossRefGoogle Scholar
Neumann, H., Varieties of Groups, 1st edn, Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge, 37 (Springer, Berlin, 1967).10.1007/978-3-642-88599-0CrossRefGoogle Scholar
Oates, S. and Powell, M. B., ‘Identical relations in finite groups’, J. Algebra 1 (1964), 11–39.CrossRefGoogle Scholar