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Strongly bounded automorphism groups. (English) Zbl 1098.20003

A group \(G\) is Cayley bounded if for every generating subset \(U\subset G\) there exists \(n\in\omega\) such that every element of \(G\) is a product of \(n\) elements of \(U\cup U^{-1}\cup\{1\}\). A group is strongly bounded if it is Cayley bounded and has cofinality strictly greater than \(\omega\).
In this paper the author shows that the automorphism groups of typical countable structures with the small index property are strongly bounded. In particular, it is proved that this is the case when \(G\) is the automorphism group of the countable universal locally finite extension of a periodic Abelian group.

MSC:

20B27 Infinite automorphism groups
20F50 Periodic groups; locally finite groups
20F05 Generators, relations, and presentations of groups
03C60 Model-theoretic algebra
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