The linear part of a discontinuously acting Euclidean semigroup

Let ℝⁿ be a Euclidean space and let S be a Euclidean semigroup, i.e., a subsemigroup of the group of isometrics of ℝⁿ. We say that a semigroup S acts discontinuously on ℝⁿ if the subset {s S : sK ∩ K ≠ ∅} is finite for any compact set K of ℝⁿ. The main results of this work are Theorem. If S is a Euclidean semigroup which acts discontinuously on ℝⁿ, then the connected component of the closure of the linear part ℓ(S) of S is a reducible group. Corollary. Let S be a Euclidean semigroup acting discontinuously on ℝⁿ; then the linear part ℓ(S) of S is not dense in the orthogonal group O(n). These results are the first step in the proof of the following Margulis' Conjecture. If S is a crystallographic Euclidean semigroup, then S is a group.