Vol. 44, No. 2, 1973

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 332: 1
Vol. 331: 1  2
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
On groups of exponent four satisfying an Engel condition

R. B. Quintana and Charles R. B. Wright

Vol. 44 (1973), No. 2, 701–705
Abstract

Let B(n) be the Burnside (i.e., freest) group of exponent 4 on n generators. It is known that B(n) is nilpotent of class at most 3n 1. This paper exhibits a commutator of length 3n 1 in B(n) which must be nontrivial if the class is exactly 3n 1. The methods also yield an easy proof of the following.

Theorem. Let E(n) be B(n) reduced modulo the identical commutator relation

(a1,⋅⋅⋅ ,a2n−4,x,x,(y,z,z,z)) = 1.

Then E(n) is nilpotent of class at most 2n + 3.

As an immediate corollary, every n-generator group of exponent 4 satisfying the Engel condition (x,y,y,y) = 1 identically is of class at most 2n + 3.

Mathematical Subject Classification 2000
Primary: 20F45
Milestones
Received: 1 October 1971
Revised: 10 January 1972
Published: 1 February 1973
Authors
R. B. Quintana
Charles R. B. Wright