The group of all
automorphisms of a chain Ω forms a lattice-ordered group A(Ω) under the pointwise
order. It is well known that if G is the symmetric group on ℵ elements (ℵ≠6), then
every automorphism of G is inner. Here it is shown that if Ω is an α-set, every
l-automorphism of A(Ω) (preserving also the lattice structure) is inner. This is
accomplished by means of an investigation of the orbits ωA(Ω) of Dedekind cuts ω of
Ω.
The same conjecture for arbitrary chains Ω has been investigated in [6], [4], and
[8]. Lloyd proved in [6] that when Ω is the chain of rational numbers (i.e., the 0-set),
or is Dedekind complete, every l-automorphism of A(Ω) is inner. He also stated this
conclusion for α-sets in general, but a lacuna in his proof has been pointed out by C.
Holland.
|