Greg Egan on Nostr: I should just add that, while the equation: D(a c) = a D(c) + D(a) makes the map: x ...

I should just add that, while the equation:

D(a c) = a D(c) + D(a)

makes the map:

x ↦ a x + b
↓
x ↦ a x + (b q + D(a))

into an automorphism (where q is any invertible element), when you *also* need it to be class-preserving and outer, if you compare this with what happens to

x ↦ a x + b

conjugated by:

x ↦ c x + d

Namely:

x ↦ a x + (b c + d (1-a))

you have the additional constraint that:

D(a) = δ(a) (1-a)

where δ(a) is not a constant function.