9. INNER AMENABILITY 59
countable group which is perfect, i.e., every automorphism is inner. On the
other hand, it is well-known that if Γ is infinite abelian, then Γ is a sub-
group of a compact abelian Polish group and thus a non-discrete normal
subgroup of a Polish group. Indeed, if Γ is an infinite abelian group and
ˆ
Γ
is its dual, there is a countable subset I
ˆ
Γ which separates points (i.e.,
∀γ Γ \ {1}∃ˆ γ I(ˆ(γ) γ = 1)). Then consider the (compact abelian Polish)
product group G =
TI
and embed (algebraically) Γ into G via π(γ) = γ|I
(where we view here γ as a character of
ˆ
Γ).
In the reverse direction, it is also not clear what non-discrete Polish
groups admit a dense normal subgroup. Clearly every non-discrete abelian
Polish group and the infinite symmetric group have this property.
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