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.