  Surprisingly, an interesting property of the Noether charge that it is by
itself invariant under the corresponding symmetry transformation is never
discussed in quantum field theory or classical mechanics textbooks we have
checked. This property is also almost never mentioned in articles devoted to
Noether's theorem. Nevertheless, to prove this property in the context of
Lagrangian formalism is not quite trivial and the proof, outlined in this
article, can constitute an useful and interesting exercise for students.
