Abstract
It is proved that if V(X) is a proper verbal subgroup of a free group X of countable rank, then a verbal subgroup V(H) of the complete direct product\(H = \tilde \Pi \times _{X_i } \) of a countable number of isomorphic copies Xi of X differs from the complete direct product\(\tilde \Pi \times _{V_i } (X_i )\) of copies of the verbal subgroup of the factors.
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A. Steinberg and G. Baumslag, “Residual nilpotence and relations in free groups,” Bull. Amer. Math., Soc.,70, 283–284 (1964).
G. Baumslag, “Residual nilpotence and relations in free groups,” J. of Algebra,2, No. 3, 271–282 (1965).
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Translated from Matematicheskie Zametki, Vol. 9, No. 6, pp. 687–692, June, 1971.
The author wishes to thank O. N. Golovin for suggesting this problem.
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Ashmanov, S.A. Verbal subgroups of complete direct products of groups. Mathematical Notes of the Academy of Sciences of the USSR 9, 399–401 (1971). https://doi.org/10.1007/BF01094583
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DOI: https://doi.org/10.1007/BF01094583