What do these definitions of conjugacy have in common?

Question

Here are four (seemingly) different uses of the word conjugate:

1. Complex conjugates are a concrete instance of the idea of conjugacy in field extensions.

2. In group theory, there's the idea of conjugacy classes

3. In probability theory, there are conjugate distributions

4. Then there's also the convex conjugate of a function

What do they have in common? What is the most general idea of a conjugate?

Answer 1

Galois Theory says that conjugate subfields correspond to conjugate subgroups, where conjugate subfields are as in field extensions and conjugate subgroups are as in group theory.