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Some simple locally (soluble-by-finite) groups. (English) Zbl 1201.20025

Bianchi, Mariagrazia (ed.) et al., Ischia group theory 2008. Proceedings of the conference in group theory, Naples, Italy, April 1–4, 2008. Hackensack, NJ: World Scientific (ISBN 978-981-4277-79-2/pbk). 79-89 (2009).
This paper is a report on a conference talk and is in the nature of a survey with a few extra examples. The context is the following. Let \(\mathbf P\) and \(\mathbf Q\) be (presumably closely related) group theoretic properties. Given a group satisfying \(\mathbf P\) can \(G\) be embedded into a simple group satisfying \(\mathbf Q\) and, as an offshoot, how do we construct simple groups satisfying \(\mathbf Q\)? After surveying earlier results from various authors, the authors then concentrate on their more recent work in this area. We conclude here by picking out a few of their more recent results.
Every simple locally (soluble-by-finite) group is locally residually finite. Let \(G\) be a residually finite, countable locally (soluble-by-finite) group. Then \(G\) can be embedded into a countable simple locally (soluble-by-finite) group. For each positive integer \(c\) there exists a countable simple group that is locally (nilpotent of class at most \(c\))-by-finite but not locally (nilpotent of class at most \(c-1\))-by-finite.
For the entire collection see [Zbl 1176.20001].

MSC:

20E32 Simple groups
20E25 Local properties of groups
20E26 Residual properties and generalizations; residually finite groups
20F19 Generalizations of solvable and nilpotent groups
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