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.


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


Zbl 1192.20017
Full Text: DOI EuDML Link


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.