Normal closure of one-variable equations in free groups

Author:
C. Sibertin-Blanc

Journal:
Proc. Amer. Math. Soc. **80** (1980), 34-38

MSC:
Primary 20E05

DOI:
https://doi.org/10.1090/S0002-9939-1980-0574504-2

MathSciNet review:
574504

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a one-variable equation in a free group *F* of finite rank. Lyndon has proved that it is possible to associate effectively to the set of its solutions, whereas Appel and Lorenc have provided a simpler representation of the set inferred. In this paper, we invert the problem and demonstrate that if the elements of any set are solutions of an equation , then belongs to the normal closure of finitely many short equations associated to *S*. A few consequences are given.

**[1]**K. I. Appel,*One-variable equations in free groups*, Proc. Amer. Math. Soc.**19**(1968), 912-919. MR**0232826 (38:1149)****[2]**A. A. Lorencs,*Representations of sets of solutions of systems of equations with one unknown in a free group*, Dokl. Akad. Nauk SSSR**178**(1968), 290-292. MR**0225861 (37:1452)****[3]**R. C. Lyndon,*Equations in free groups*, Trans. Amer. Math. Soc.**96**(1960), 445-457. MR**0151503 (27:1488)****[4]**R. C. Lyndon and P. E. Schupp,*Combinatorial group theory*, Springer-Verlag, Berlin, 1977. MR**0577064 (58:28182)****[5]**A. Steinberg,*On equations in free groups*, Michigan Math. J.**18**(1971), 87-95. MR**0289614 (44:6802)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0574504-2

Keywords:
One-variable equation,
set of solutions,
free generator,
parametric word,
normal closure,
primitive element,
recursively enumerable

Article copyright:
© Copyright 1980
American Mathematical Society