  In this article we observe that a locally compact group $G$ is completely
determined by the algebraic properties of its Feichtinger's Segal algebra
$S_0(G).$ Let $G$ and $H$ be locally compact groups. Then any linear (not
necessarily continuous) bijection of $S_0(G)$ onto $S_0(H)$ which preserves the
convolution and pointwise products is essentially a composition with a
homeomorphic isomorphism of $H$ onto $G.$
