Crystallographic properties of local groups of a Delone set in a Euclidean plane

It is proved that, in any Delone set on a Euclidean plane, a subset of points with a crystallographic local group, i.e., with local rotations of order $n$ = 1,2,3,4, or 6, is also a Delone set. This result has a number of important implications for regular systems and crystalline structures. By the local group at a point of a set $X$, we mean the group of the cluster of radius 2$R$ centered at this point, where $R$ is the radius of a covering of the plane by equal disks with centers in $X$.