TY - JOUR

T1 - A note on conciseness of Engel words

AU - Fernandez-Alcober, Gustavo A

AU - Moriagi, Marta

AU - Traustason, Gunnar

PY - 2012

Y1 - 2012

N2 - It is still an open problem to determine whether the n-th
Engel word [x,_n y] is concise, that is, if for every group G such that
the set of values e_n(G) taken by [x,_n y] on G is finite it follows that
the verbal subgroup E_n(G) generated by e_n(G) is also finite. We prove
that if e_n(G) is finite then [En_(G),G] is finite, and either G=[E_n(G),G]
is locally nilpotent and E_n(G) is finite, or G has a finitely generated
section that is an infinite simple n-Engel group. It follows that [x_n y]
is concise if n is at most four.

AB - It is still an open problem to determine whether the n-th
Engel word [x,_n y] is concise, that is, if for every group G such that
the set of values e_n(G) taken by [x,_n y] on G is finite it follows that
the verbal subgroup E_n(G) generated by e_n(G) is also finite. We prove
that if e_n(G) is finite then [En_(G),G] is finite, and either G=[E_n(G),G]
is locally nilpotent and E_n(G) is finite, or G has a finitely generated
section that is an infinite simple n-Engel group. It follows that [x_n y]
is concise if n is at most four.

UR - http://www.scopus.com/inward/record.url?scp=84863826067&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1080/00927872.2011.582061

U2 - 10.1080/00927872.2011.582061

DO - 10.1080/00927872.2011.582061

M3 - Article

VL - 40

SP - 2570

EP - 2576

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 7

ER -