On model-theoretic connected components in some group extensions

Details On model-theoretic connected components in some group extensions On model-theoretic connected components in some group extensions

2012-01-25

arxiv.org/abs/1201.5221v1

Mathematics

Logic

Scientific paper

Assume G is a group acting by automorphisms on an abelian group A and that all the data is definable in a first order structure \G. Suppose h: G x G -> A is a B-definable (in \G) 2-cocycle with finite image contained in B (for some finite set B) and G is the corresponding extension of G by A. Let \G* be a monster model of Th(\G) and G* the interpretation of G in \G*. We prove that under some general hypothesis, G*~000_B \ne G*~00_B, where G*~00_B is the smallest B-type-definable in \G* subgroup of G* of bounded index and G*~000_B is the smallest invariant under Aut(\G*/B) subgroup of G* of bounded index. We apply this theorem to produce new classes of examples of groups for which the smallest B-type-definable subgroup of bounded index differs from the smallest B-invariant subgroup of bounded index.

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