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Embedding into groups with well-described lattices of subgroups. (English) Zbl 0864.20018

The author suggests a new scheme of embedding an arbitrary set of groups into an infinite simple group with a ‘well-described’ lattice of subgroups and a prescribed outer automorphism group. The proof is based on the techniques developed by A. Yu. Ol’shanskij [Geometry of defining relations in groups (1991; Zbl 0676.20014)] and heavily depends on some previous papers of the author and Ol’shanskij. One of the applications of the scheme is a construction (assuming the Continuum Hypothesis) of an uncountable group such that (1) every subgroup is countable, and (2) it embeds every countable group. This improves Shelah’s result who constructed an uncountable group with (1).

MSC:

20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
20F06 Cancellation theory of groups; application of van Kampen diagrams
20E15 Chains and lattices of subgroups, subnormal subgroups
20E32 Simple groups

Citations:

Zbl 0676.20014
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References:

[1] DOI: 10.1016/S0049-237X(08)71346-6 · doi:10.1016/S0049-237X(08)71346-6
[2] Ol’shanskii, Geometry of defining relations in groups (1989)
[3] Ol’shanskii, Vestnik Moskovsk. Univ. 2 pp 28– (1989)
[4] DOI: 10.1112/jlms/s1-24.4.247 · Zbl 0034.30101 · doi:10.1112/jlms/s1-24.4.247
[5] DOI: 10.1080/00927879408825167 · Zbl 0836.20033 · doi:10.1080/00927879408825167
[6] Obraztsov, Mat. Sb. 180 pp 529– (1989)
[7] DOI: 10.1112/jlms/s1-12.46.120 · Zbl 0016.29501 · doi:10.1112/jlms/s1-12.46.120
[8] Obraztsov, J. Austral. Math. Soc.
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