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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Finite nilpotent characteristic nonverbal groups

Author: Orin Chein
Journal: Trans. Amer. Math. Soc. 148 (1970), 533-548
MSC: Primary 20.10
MathSciNet review: 0257194
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Abstract: In this paper, we study nilpotent groups which are quotient groups of finitely generated free groups with respect to characteristic but nonverbal subgroups. We show that there are no abelian groups of the type in question. We also show that all such groups of nilpotence class 2 or 3 are finite and have minimal sets of two generators. In fact, formal presentations for all such groups are given.

The direct product of two finite CNV groups (as the groups in question will be called) which have minimal sets of generators of the same size is shown to again be a CNV group, provided that the orders of the original two groups are relatively prime. Conversely, if a finite CNV group is a direct product of groups of relatively prime orders, then at least one of these direct factors is a CNV group. Several other related results are also obtained.

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Keywords: Free group, characteristic subgroup, fully invariant subgroup, verbal subgroup, quaternions, nilpotent group, abelian group, finite group, minimal set of generators, direct product, direct factors, quotient group, Nielsen transformations, commutator, relation, presentation, Witt-Hall identities, P-group, nilpotence class
Article copyright: © Copyright 1970 American Mathematical Society