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On three-Engel groups

Published online by Cambridge University Press:  17 April 2009

L.-C. Kappe  and
W.P. Kappe
L.-C. Kappe
Affiliation:
Department of Mathematics, State University of New York at Binghamton, New York, USA.
W.P. Kappe
Affiliation:
Department of Mathematics, State University of New York at Binghamton, New York, USA.
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Abstract

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The following conditions for a group G are investigated:

(i) maximal class n subgroups are normal,

(ii) normal closures of elements have nilpotency class n at most,

(iii) normal closures are n–Engel groups,

(iv) G is an (n+1 )-Engel group.

Each of these conditions is a consequence of the preceding one. The second author has shown previously that all conditions are equivalent for n = 1. Here the question is settled for n = 2 as follows: conditions (ii), (iii) and (iv) are equivalent. The class of groups defined by (i) is not closed under homomorphisms, and hence (i) does not follow from the other conditions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 7 , Issue 3 , December 1972 , pp. 391 - 405
Copyright
Copyright © Australian Mathematical Society 1972

References

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