  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
isomorphism form a semigroup, denoted by $T(\pi)$. We will give a complete
proof of the following theorem due to Endo and Miyata \cite{EM5}. 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 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 $m$).
