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An Aschbacher-O’Nan-Scott theorem for countable linear groups. (English) Zbl 1285.20002

Let \(\Gamma\) be a countable linear group that admits a faithful primitive action on a set. If \(\Gamma\) is of affine or diagonal type, then this primitive action is uniquely determined (up to isomorphism), [see T. Gelander and Y. Glasner, Geom. Funct. Anal. 17(2007), No. 5, 1479-1523 (2008; Zbl 1138.20005)]. In the remaining case, \(\Gamma\) is of almost simple type (in the sense of algebraic groups). For this case, the authors construct uncountably many non-isomorphic faithful primitive actions, using a construction from loc. cit.

MSC:

20B07 General theory for infinite permutation groups
20G15 Linear algebraic groups over arbitrary fields
20B15 Primitive groups

Citations:

Zbl 1138.20005
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