  It is known that a lattice simplex of dimension $d$ corresponds a finite
abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$. Conversely, given a finite
abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$ such that the sum of all
entries of each element is an integer, we can obtain a lattice simplex of
dimension $d$. In this paper, we discuss a characterization of Gorenstein
simplices in terms of the associated finite abelian groups. In particular, we
present complete characterizations of Gorenstein simplices whose normalized
volume equals $p,p^2$ and $pq$, where $p$ and $q$ are prime numbers with $p
\neq q$. Moreover, we compute the volume of the dual simplices of Gorenstein
simplices.
