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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Isomorphism types in wreath products and effective embeddings of periodic groups

Authors: Kenneth K. Hickin and Richard E. Phillips
Journal: Trans. Amer. Math. Soc. 277 (1983), 765-778
MSC: Primary 20E22; Secondary 03D40, 20F10, 20F50
MathSciNet review: 694387
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Abstract: For any finitely generated group $ Y,\omega (Y)$ denotes the Turing degree of the word problem of $ Y$. Let $ G$ be any non-Abelian $ 2$-generator group and $ B$ an infinite group generated by $ k \geqslant 1$ elements. We prove that if $ \tau $ is any Turing degree with $ \tau \geqslant 1.{\text{u.b.}}\{ {\omega (G),\omega (B)} \}$ then the unrestricted wreath product $ W = G{\text{Wr}}\,B$ has a $ ( {k + 1} )$-generator subgroup $ H$ with $ \omega (H) = \tau $. If $ B$ is also periodic, then $ W$ has a $ k$-generator subgroup $ H$ such that $ \tau = 1.{\text{u.b.}}\{ {\omega (B),\omega (H)} \}$.

Easy consequences include: $ G{\text{Wr}}\,{\mathbf{Z}}$ has $ {2^{{\aleph _0}}}$ pairwise nonembeddable $ 2$-generator subgroups and if $ B$ is periodic then $ G{\text{Wr}}\,B$ has $ {2^{{\aleph _0}}}$ pairwise nonembeddable $ k$-generator subgroups.

Using similar methods, we prove an effective embedding theorem for embedding countable periodic groups in $ 2$-generator periodic groups.

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Keywords: Wreath product, nonembeddable finitely generated groups, periodic groups, effective embedding, word problem
Article copyright: © Copyright 1983 American Mathematical Society