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Simple groups with prescribed local structure. (English) Zbl 1229.20024

In this paper the author constructs simple groups of certain types. In particular let \(\mathcal B\) be a variety in which the free groups of infinite rank are residually finite. The author shows that for every cardinal \(\kappa\) there exists a locally (\(\mathcal B\)-by-finite) simple group \(S(\kappa)\) of cardinality equal to \(\max\{\kappa,\aleph_0\}\) which contains a copy of the \(\mathcal B\)-free group of rank \(\kappa\). This generalizes a theorem of M. R. Dixon, M. J. Evans and H. Smith [J. Group Theory 12, No. 5, 745-752 (2009; Zbl 1192.20017)]. This is a very nice paper.

MSC:

20E32 Simple groups
20E25 Local properties of groups
20F50 Periodic groups; locally finite groups
20E26 Residual properties and generalizations; residually finite groups

Citations:

Zbl 1192.20017
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References:

[1] F. Del Castillo, Modular permutations on \({Z}\), Rend. Sem. Mat. Univ. Padova, 118 (2007), pp. 147–158. Zbl1166.20002 MR2378393 · Zbl 1166.20002
[2] M. R. Dixon - M. J. Evans - H. Smith, Embedding groups in locally (soluble-by-finite) simple groups, J. Group Theory, 9 (2006), pp. 383–395. Zbl1120.20030 MR2226620 · Zbl 1120.20030 · doi:10.1515/JGT.2006.026
[3] M. R. Dixon - M. J. Evans - H. Smith, Simple groups with prescribed local properties, J. Group Theory, 12 (2009), pp. 745–752. Zbl1192.20017 MR2554765 · Zbl 1192.20017 · doi:10.1515/JGT.2009.008
[4] M. R. Dixon - M. J. Evans - H. Smith, Some simple locally (soluble-by-finite) groups, pp. 79–89 in Ischia Group Theory 2008, M. Bianchi, P. Longobardi, M. Maj and C. M. Scoppola (eds.), World Scientific (2009). Zbl1201.20025 · Zbl 1201.20025
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