VARIETIES OF GROUPS OF EXPONENT 4

It is proved that the variety of all 4-Engel groups of exponent 4 is a maximal proper subvariety of the Burnside variety [Bfr ]4, and the consequences of this are discussed for the finite basis problem for varieties of groups of exponent 4. It is proved that, for r [ges ] 2, the 4-Engel verbal subgr...

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Bibliographic Details
Published in:Journal of the London Mathematical Society 1999-12, Vol.60 (3), p.747-756
Main Author: QUICK, MARTYN
Format: Article
Language:English
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Summary:It is proved that the variety of all 4-Engel groups of exponent 4 is a maximal proper subvariety of the Burnside variety [Bfr ]4, and the consequences of this are discussed for the finite basis problem for varieties of groups of exponent 4. It is proved that, for r [ges ] 2, the 4-Engel verbal subgroup of the r-generator Burnside group B(r, 4) is irreducible as an [ ]2GL(r, 2)-module. It is observed that the variety of all 4-Engel groups of exponent 4 is insoluble, but not minimal insoluble.
ISSN:0024-6107
1469-7750
DOI:10.1112/S0024610799008169