Bogomolov multipliers and retract rationality for semi-direct products
Abstract
Let
G
be a finite group. The Bogomolov multiplier
B
0
(G)
is constructed as an obstruction to the rationality of $\bm{C}(V)^G$ where
G→GL(V)
is a faithful representation over $\bm{C}$. We prove that, for any finite groups
G
1
and
G
2
,
B
0
(
G
1
×
G
2
)
−
→
∼
B
0
(
G
1
)×
B
0
(
G
2
)
under the restriction map. If
G=N⋊
G
0
with
gcd{|N|,|
G
0
|}=1
, then
B
0
(G)
−
→
∼
B
0
(N
)
G
0
×
B
0
(
G
0
)
under the restriction map. For any integer
n
, we show that there are non-direct-product
p
-groups
G
1
and
G
2
such that
B
0
(
G
1
)
and
B
0
(
G
2
)
contain subgroups isomorphic to $(\bm{Z}/p \bm{Z})^n$ and $\bm{Z}/p^n \bm{Z}$ respectively. On the other hand, if
k
is an infinite field and
G=N⋊
G
0
where
N
is an abelian normal subgroup of exponent
e
satisfying that
ζ
e
∈k
, we will prove that, if
k(
G
0
)
is retract
k
-rational, then
k(G)
is also retract
k
-rational provided that certain "local" conditions are satisfied; this result generalizes two previous results of Saltman and Jambor \cite{Ja}.