  In the paper, we prove that in an arbitrary Delone set $X$ in $3D$ space, the
subset $X_6$ of all points from $X$ at which local groups have axes of the
order not greater than 6 is also a Delone set. Here, under the local group at
point $x\in X$ is meant the symmetry group $S_x(2R)$ of the cluster $C_x(2R)$
of $x$ with radius $2R$, where $R$ (according to Delone's theory of the 'empty
sphere') is the radius of the largest 'empty' ball, that is, the largest ball
free of points of $X$.
  The main result seems to be the first rigorously proved statement on
absolutely generic Delone sets which implies substantial statements for Delone
sets with strong crystallographic restrictions. For instance, an important
observation of Shtogrin on the boundedness of local groups in Delone sets with
equivalent $2R$-clusters immediately follows from the main theorem.
  In the paper, the 'crystalline kernel conjecture' (Conjecture 1) and its two
weaker versions (Conjectures 2 and 3) are suggested. According to Conjecture 1,
in a quite arbitrary Delone set, points with locally crystallographic axes (of
order 2,3,4, or 6) only inevitably constitute an essential part of the set.
These conjectures significantly generalize the famous statement of
Crystallography on the impossibility of (global) 5-fold symmetry in a 3D
lattice.
