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This article studies the group generated by automorphisms of the projectivespace of dimension $n$ and by the standard birational involution of degree $n$.Every element of this group only contracts rational hypersurfaces, but in odddimension, there are simple elements having this property which do not belongto the group. Geometric properties of the elements of the group are given, aswell as a description of its intersection with monomial transformations.

The group of Cremona transformations generated by linear maps and the standard involution