  In the paper the foundation of the $k$-orbit theory is developed. The theory
opens a new simple way to the investigation of groups and multidimensional
symmetries.
  The relations between combinatorial symmetry properties of a $k$-orbit and
its automorphism group are found. It is found the local property of a
$k$-orbit. The difference between 2-closed group and $m$-closed group for $m>2$
is discovered. It is explained the specific property of Petersen graph
automorphism group $n$-orbit. It is shown that any non-trivial primitive group
contains a transitive imprimitive subgroup and as a result it is proved that
the automorphism group of a vertex transitive graph (2-closed group) contains a
regular element (polycirculant conjecture).
  Using methods of the $k$-orbit theory, it is considered different
possibilities of permutation representation of a finite group and shown that
the most informative, relative to describing of the structure of a finite
group, is the permutation representation of the lowest degree. Using this
representation it is obtained a simple proof of the W. Feit, J.G. Thompson
theorem: Solvability of groups of odd order. It is described the enough simple
structure of lowest degree representation of finite groups and found a way to
constructing of the simple full invariant of a finite group.
  To the end, using methods of $k$-orbit theory, it is obtained one of possible
polynomial solutions of the graph isomorphism problem.
