Let $K/k$ be a finite Galois extension
and $\pi = \fn{Gal}(K/k)$. An algebraic torus $T$ defined over $k$
is called a $\pi$-torus if $T\times_{\fn{Spec}(k)}
\fn{Spec}(K)\simeq \bm{G}_{m,K}^n$ for some integer $n$. The set
of all algebraic $\pi$-tori defined over $k$ under the stably birational
equivalence forms a semigroup, denoted by $T(\pi)$. We will give a
complete proof of the following theorem due to Endo and Miyata
\cite{EM4}. Theorem. Let $\pi$ be a finite group. Then
$T(\pi)\simeq C(\Omega_{\bm{Z}\pi})$ where $\Omega_{\bm{Z}\pi}$ is
a maximal $\bm{Z}$-order in $\bm{Q}\pi$ containing $\bm{Z}\pi$ and
$C(\Omega_{\bm{Z}\pi})$ is the locally free class group of
$\Omega_{\bm{Z}\pi}$, provided that $\pi$ is isomorphic to one of the
following four types of groups : $C_n$ ($n$ is any positive
integer), $D_m$ ($m$ is any odd integer $\ge 3$), $C_{q^f}\times
D_m$ ($m$ is any odd integer $\ge 3$, $q$ is an odd prime number
not dividing $m$, $f\ge 1$, and
$(\bm{Z}/q^f\bm{Z})^{\times}=\langle \bar{p}\rangle$ for any prime
divisor $p$ of $m$), $Q_{4m}$ ($m$ is any odd integer $\ge 3$,
$p\equiv 3 \pmod{4}$ for any prime divisor $p$ of $m$).