  We show that two ordinary isogenous elliptic curves have isomorphic groups of
rational points if they have the same $j$-invariant and we extend this result
to certain isogenous supersingular elliptic curves, namely those with equal
$j$-invariant of either 0 or 1728. Using a result by Heuberger and Mazzoli we
establish a general case of this relationship within isogenous elliptic curves
not necessarily having equal $j$-invariant.
