Vol. 44, No. 2, 1973

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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