×

Permutation groups containing infinite symplectic linear groups and reducts of linear spaces over the two element field. (English) Zbl 1388.20005

Summary: Let \(\mathbb{F}_2^\omega\) denote the countably infinite dimensional vector space over the two element field and \(\mathrm{GL}(\omega,2)\) its automorphism group. Let \(\operatorname{Sym}(\mathbb{F}_2^\omega)\) denote the symmetric group acting on the elements of \(\mathbb{F}_2^\omega\) and \(\mathrm{Sp}\) denote the symplectic group. The closed intermediate subgroups \(G\), such that \(\mathrm{Sp}\leq G\leq \operatorname{Sym}(\mathbb{F}_2^\omega)\) are characterized. As \(\mathbb{F}_2^\omega\) endowed with the standard symplectic form is an \(\omega\)-categorical structure, these groups correspond to its first order definable reducts. These reducts are also analyzed.

MSC:

20B27 Infinite automorphism groups
20B35 Subgroups of symmetric groups
03C05 Equational classes, universal algebra in model theory
03C35 Categoricity and completeness of theories
03C60 Model-theoretic algebra
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/BF01234360 · Zbl 0453.20004
[2] DOI: 10.2178/jsl/1286198146 · Zbl 1408.03021
[3] DOI: 10.1007/s11856-014-1042-y · Zbl 1292.05231
[4] Bodirsky M., Contemp. Math. 558 pp 489– (2011)
[5] DOI: 10.2178/jsl.7804020 · Zbl 1327.03008
[6] Bogomolov F., C. Euro J. Math. 11 pp 17– (2013)
[7] DOI: 10.1007/BF01214702 · Zbl 0313.20022
[8] Fraïssé R., Comptes Rendus de l’Académie des Sci. de Paris 237 pp 540– (1953)
[9] DOI: 10.1017/CBO9780511551574
[10] DOI: 10.2178/jsl/1230396752 · Zbl 1189.03041
[11] DOI: 10.1112/jlms/s2-8.3.426 · Zbl 0299.20010
[12] DOI: 10.1090/tran/6608 · Zbl 1397.20007
[13] DOI: 10.1016/j.disc.2011.01.024 · Zbl 1238.03032
[14] DOI: 10.1016/j.aim.2014.08.008 · Zbl 1403.03053
[15] DOI: 10.1016/j.jcta.2013.04.003 · Zbl 1318.06001
[16] Pogorelov P. A., Mat. Zametki 16 pp 91– (1974)
[17] Pongrácz A., Ann. Pure Appl. Logic
[18] Thomas S., Reducts of the random graph 56 (1) pp 176– (1991) · Zbl 0743.05049
[19] DOI: 10.1016/0168-0072(95)00061-5 · Zbl 0865.03025
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.