This is a very easy fact we use in Group Theory,
But somehow, I wondered that whether there may be another way (other than Lagrange's Theorem) to prove that the order of an element divides the order of Group.
I attempted to go on the term "exponent" of the Group (just assume that G is a Finite Group), which we may define the least common multiple of orders of elements in G. But this exponent divides $|G|$, since the order of each element divides $|G|$. So it became a little paradox.
Shall I think that; before teaching this fact in a Course, one should teach about Lagrange's first?