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Summary: BOUNDED GENERATION AND LINEAR GROUPS
MIKL
OS AB
ERT, ALEXANDER LUBOTZKY, AND L
ASZL
O PYBER
Abstract. A group is called boundedly generated (BG) if it is the set-
theoretic product of nitely many cyclic subgroups. We show that a BG group
has only abelian by nite images in positive characteristic representations.
We use this to reprove and generalise Rapinchuk's theorem by showing
that a BG group with the FAb property has only nitely many irreducible
representations in any given dimension over any eld. We also give a structure
theorem for the pro nite completion G of such a group .
On the other hand, we exhibit boundedly generated pro nite FAb groups
which do not satisfy this structure theorem.
1. Introduction
A group (resp. pro nite group) is said to have bounded generation (or nite
cyclic width) if is a product of its cyclic (resp. procyclic) subgroups C 1 ; : : : ; C k .
The smallest number k for which has such a decomposition is called the cyclic
width of (which we denote by cw()).
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