Following Procesi and Formanek, the center of the division ring of
n
×
n
n\times n
generic matrices over the complex numbers
C
\mathbf C
is stably equivalent to the fixed field under the action of
S
n
S_n
, of the function field of the group algebra of a
Z
S
n
ZS_n
-lattice, denoted by
G
n
G_n
. We study the question of the stable rationality of the center
C
n
C_n
over the complex numbers when
n
n
is a prime, in this module theoretic setting. Let
N
N
be the normalizer of an
n
n
-sylow subgroup of
S
n
S_n
. Let
M
M
be a
Z
S
n
ZS_n
-lattice. We show that under certain conditions on
M
M
, induction-restriction from
N
N
to
S
n
S_n
does not affect the stable type of the corresponding field. In particular,
C
(
G
n
)
\mathbf C (G_n)
and
C
(
Z
G
⊗
Z
N
G
n
)
\mathbf C(ZG\otimes _{ZN}G_n)
are stably isomorphic and the isomorphism preserves the
S
n
S_n
-action. We further reduce the problem to the study of the localization of
G
n
G_n
at the prime
n
n
; all other primes behave well. We also present new simple proofs for the stable rationality of
C
n
C_n
over
C
\mathbf C
, in the cases
n
=
5
n=5
and
n
=
7
n=7
.