  The symmetry study of main differential equations of mechanics and
electrodynamics has shown, that differential equations, which are invariant
under transformations of groups, which are symmetry groups of mathematical
numbers (considered within the frames of the number theory) determine the
mathematical nature of the quantities, incoming in given equations. It allowed
to proof the main postulate of quantum mechanics, consisting in that, that to
any mechanical quantity can be set up into the correspondence the Hermitian
matrix by quantization.
  High symmetry of Maxwell equations allows to show, that to quantities,
incoming in given equations can be set up into the correspondence the
Quaternion (twice-Hermitian) matrix by their quantization.
  It is concluded, that the equations of the dynamics of mechanical systems are
not invariant under transformations of quaternion multiplicative group and,
consecuently, direct application of quaternions with usually used basis \{e, i,
j, k \} to build the new version of quantum mechanics, which was undertaken in
the number of modern publications, is incorrect. It is the consequence of
non-abelian character of given group. At the same time we have found the
correct ways for the creation of the new versions of quantum mechanics on the
quaternion base by means of choice of new bases in quaternion ring, from which
can be formed the bases for complex numbers under multiplicative groups of
which the equations of the dynamics of mechanical systems are invariant.
