Infinite cyclic verbal subgroups

of relatively free groups

Author:
A. Storozhev

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2953-2954

MSC (1991):
Primary 20E10, 20F06

DOI:
https://doi.org/10.1090/S0002-9939-96-03521-6

MathSciNet review:
1343726

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Abstract: We prove that there exist a relatively free group and a word in two variables such that the verbal subgroup of defined by is an infinite cyclic group whereas has only one nontrivial value in .

**[1]**S.V. Ivanov,*P. Hall's conjecture on the finiteness of verbal subgroups*, Izv. Vyssh. Ucheb. Zaved.**325**(1989), 60--70. MR**90j:20061****[2]**Kourovka Notebook,*Unsolved problems of the group theory*, Tenth Edition, Novosibirsk, 1986.**[3]**Kourovka Notebook,*Unsolved problems of the group theory*, Eleventh Edition, Novosibirsk, 1991.**[4]**A.Yu. Ol'shanskii,*Geometry of defining relations in groups*, Mathematics and Its Applications (Soviet Series), vol. 70, Kluwer Academic Publishers, Dordrecht, 1991. MR**93g:20071****[5]**A. Storozhev,*On abelian subgroups of relatively free groups*, Comm. Algebra**22**(1994), 2677--2701. MR**95d:20066****[6]**R.F. Turner-Smith,*Marginal subgroup properties for outer commutator words*, Proc. London. Math. Soc.**14**(1964), 321--341. MR**29:2289**

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Additional Information

**A. Storozhev**

Affiliation:
Australian Mathematics Trust, University of Canberra, PO Box 1, Belconnen, ACT 2616, Australia

Email:
ans@amt.canberra.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-96-03521-6

Received by editor(s):
March 6, 1995

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1996
American Mathematical Society