Neumann, Peter M.; Praeger, Cheryl E.; Smith, Simon M. Some infinite permutation groups and related finite linear groups. (English) Zbl 1430.20003 J. Aust. Math. Soc. 102, No. 1, 136-149 (2017). Summary: This article began as a study of the structure of infinite permutation groups \(G\) in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy \(\min\)-\(n\), the minimal condition on normal subgroups. The groups \(G\) are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup \(M\) which is a divisible abelian \(p\)-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a \(p\)-adic vector space associated with \(M\). This leads to our second variation, which is a study of the finite linear groups that can arise. Cited in 1 Document MSC: 20B07 General theory for infinite permutation groups 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20C10 Integral representations of finite groups 20C20 Modular representations and characters Keywords:infinite permutation groups; finite groups; modules; modular representations PDFBibTeX XMLCite \textit{P. M. Neumann} et al., J. Aust. Math. Soc. 102, No. 1, 136--149 (2017; Zbl 1430.20003) Full Text: DOI arXiv References: [1] George M.Bergman and Henrik W.LenstraJr, ‘Subgroups close to normal subgroups’, J. Algebra127 (1989), 80-97.10.1016/0021-8693(89)90275-5 · Zbl 0641.20023 [2] Charles W.Curtis and IrvingReiner, Representation Theory of Finite Groups and Associative Algebras (John Wiley, New York, 1962). · Zbl 0131.25601 [3] ManfredDroste and RüdigerGöbel, ‘McLain groups over arbitrary rings and orderings’, Math. Proc. Cambridge Philos. Soc.117 (1995), 439-467.10.1017/S0305004100073291 · Zbl 0839.20050 [4] LászlóFuchs, Infinite Abelian Groups, Vol. I (Academic Press, New York, 1970). · Zbl 0209.05503 [5] P.Hall, ‘Wreath powers and characteristically simple groups’, Proc. Cambridge Philos. Soc.58 (1962), 170-184; also in Collected Works of Philip Hall (eds. K. W. Gruenberg and J. E. Roseblade) (Clarendon Press, Oxford, 1988), 611-625.10.1017/S0305004100036379 · Zbl 0109.01302 [6] GordonJames, ‘On a conjecture of Carter concerning irreducible Specht modules’, Math. Proc. Cambridge Philos. Soc.83 (1978), 11-17.10.1017/S0305004100054232 · Zbl 0385.05027 [7] GordonJames and AdalbertKerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, 16 (Addison-Wesley, Reading, MA, 1981). · Zbl 0491.20010 [8] D. H.McLain, ‘A characteristically simple group’, Proc. Cambridge Philos. Soc.50 (1954), 641-642.10.1017/S0305004100029819 · Zbl 0056.02201 [9] G.Schlichting, ‘Operationen mit periodischen Stabilisatoren’, Arch. Math. (Basel)34 (1980), 97-99.10.1007/BF01224936 · Zbl 0449.20004 [10] Simon M.Smith, ‘A classification of primitive permutation groups with finite stabilisers’, J. Algebra432 (2015), 12-21.10.1016/j.jalgebra.2015.01.023 · Zbl 1322.20005 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.