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Register a new userCartan subgroups of groups definable in o-minimal structures
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Authors
Elias Baro
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We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group, in general very largely in terms of the dimension.
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In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic $p$-group. Lastly, we note that class-preserving automorphisms of 2-power order of some nilpotent-by-nilpotent groups are inner, extending a result by J. Hai and J. Ge.
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Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing large subgroups in finite groups (see Theorems A and C). We also consider the more specialised problems of finding large (non-abelian) nilpotent as well as abelian subgroups in soluble groups.
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