Permutation Groups in o‐Minimal Structures

In this paper we develop a structure theory for transitive permutation groups definable in o‐minimal structures. We fix an o‐minimal structure M , a group G definable in M , and a set Ω and a faithful transitive action of G on Ω definable in M , and talk of the permutation group ( G , Ω). Often, we are concerned with definably primitive permutation groups ( G , Ω); this means that there is no proper non‐trivial definable G ‐invariant equivalence relation on Ω, so definable primitivity is equivalent to a point stabiliser G α being a maximal definable subgroup of G . Of course, since any group definable in an o‐minimal structure has the descending chain condition on definable subgroups [ 23 ] we expect many questions on definable transitive permutation groups to reduce to questions on definably primitive ones. Recall that a group G definable in an o‐minimal structure is said to be connected if there is no proper definable subgroup of finite index. In some places, if G is a group definable in M we must distinguish between definability in the full ambient structure M and G ‐definability, which means definability in the pure group G := ( G , .); for example, G is G ‐ definably connected means that G does not contain proper subgroups of finite index which are definable in the group structure. By definable , we always mean definability in M . In some situations, when there is a field R definable in M , we say a set is R‐semialgebraic , meaning that it is definable in (R, +, .). We call a permutation group ( G , Ω) R‐semialgebraic if G , Ω and the action of G on Ω can all be defined in the pure field structure of a real closed field R . If R is clear from the context, we also just write ‘semialgebraic’.