  Given the non-canonical relationship between variables used in the
Hamiltonian formulations of the Einstein-Hilbert action (due to Pirani, Schild,
Skinner (PSS) and Dirac) and the Arnowitt-Deser-Misner (ADM) action, and the
consequent difference in the gauge transformations generated by the first-class
constraints of these two formulations, the assumption that the Lagrangians from
which they were derived are equivalent leads to an apparent contradiction that
has been called "the non-canonicity puzzle". In this work we shall investigate
the group properties of two symmetries derived for the Einstein-Hilbert action:
diffeomorphism, which follows from the PSS and Dirac formulations, and the one
that arises from the ADM formulation. We demonstrate that unlike the
diffeomorphism transformations, the ADM transformations (as well as others,
which can be constructed for the Einstein-Hilbert Lagrangian using Noether's
identities) do not form a group. This makes diffeomorphism transformations
unique (the term "canonical" symmetry might be suggested). If the two
Lagrangians are to be called equivalent, canonical symmetry must be preserved.
The interplay between general covariance and the canonicity of the variables
used is discussed.
