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Let O be the ring of S-integers in a number field k. We prove that if the group of units O × is infinite then every matrix in Γ = SL2(O) is a product of at most 9 elementary matrices. This essentially completes a long line of research in this direction. As a consequence, we obtain a new proof of the fact that Γ is boundedly generated as an abstract group that uses only standard results from algebraic number theory. To Alex Lubotzky on his 60th birthday1 Corollary 1.2. Let O = O k,S be the ring of S-integers, in a number field k. If the group of units O × is infinite, then the group Γ = SL 2 (O) has bounded generation.We note that combining this fact with the results of [Lu], [PR1], one obtains an alternative proof of the centrality of the congruence kernel for SL 2 (O) (provided that O × is infinite), originally

In the present paper we show that comparing results of Hyman Bass [15] with those of Oleg Tavgen [10], one immediately gets the following result, which is both more general, and more precise than all recent results pertaining to unitriangular factorisations.

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We apply group actions to some natural situations like the natural 'linear' action of G L r (Z n ) and some of its subgroups to derive numbertheoretic identities like t 1 ∈(Z n ) * ,t 2 ,··· ,t r ∈Z n GC D(n, t 1 − 1, t 2 , · · · , t r ) = φ(n)σ r−1 (n).