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A classification of primitive permutation groups with finite stabilizers. (English) Zbl 1322.20005

The author extends the classical Aschbacher-O’Nan-Scott theorem to infinite primitive permutation groups which have finite point stabilizers. Specifically he proves the following theorem. Let \(G\leq\mathrm{Sym}(\Omega)\) be an infinite primitive permutation group with a finite point stabilizer \(G_\alpha\). Then \(G\) is finitely generated by elements of finite order and possesses a unique (non-trivial) minimal normal subgroup \(M=K_1\times\cdots\times K_m\) where the \(K_i\) are isomorphic infinite, nonabelian, finitely generated simple groups and such that the stabilizer \(G_\alpha\) acts transitively on \(\Gamma:=\{K_1,\ldots,K_m\}\) by conjugation. Moreover \(G\) falls into precisely one of the following categories (and there exist groups of each of the three types):
(i) \(M\) is simple and acts regularly on \(\Omega\), and \(G\) is a split extension \(M.G_\alpha\) with no non-identity element of \(G_\alpha\) inducing an inner automorphism of \(M\);
(ii) \(M\) is simple, acts non-regularly on \(\Omega\), and has finite index in \(G\), and \(M\leq G\leq\operatorname{Aut}(M)\); or
(iii) \(M\) is not simple, \(m>1\) and \(G\) is permutation isomorphic to a subgroup of the wreath product \(H\mathrm{Wr}_\Delta\mathrm{Sym}(\Delta)\) acting via the product action and \(H\) is an infinite primitive group with a finite point stabilizer (\(K\) is the unique minimal normal subgroup of \(H\)). Moreover, if \(M\) is regular, then \(H\) is of type (i) whilst if \(M\) is not regular then \(H\) is of type (ii).
Other generalizations of the classical theorem to the infinite case are given in [D. Macpherson and A. Pillay, Proc. Lond. Math. Soc., III. Ser. 70, No. 3, 481-504 (1995; Zbl 0820.03020)], [D. Macpherson and C. E. Praeger, Proc. Lond. Math. Soc., III. Ser. 68, No. 3, 518-540 (1994; Zbl 0818.20002)] and [T. Gelander and Y. Glasner, Geom. Funct. Anal. 17, No. 5, 1479-1523 (2008; Zbl 1138.20005)].

MSC:

20B07 General theory for infinite permutation groups
20B15 Primitive groups
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References:

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