  We review the depth two and Hopf algebroid-Galois theory in math.RA/0108067
and specialize to induced representations of semisimple algebras and character
theory of finite groups. We show that depth two subgroups over the complex
numbers are normal subgroups. As a converse we observe that normal Hopf
subalgebras over a field are depth two extensions. We introduce a generalized
Miyashita-Ulbrich action on the centralizer of a ring extension, and apply it
to a study of depth two and separable extensions, providing new
characterizations of separable and H-separable extensions. With a view to the
problem of when separable extensions are Frobenius, we supply a trace ideal
condition for when a ring extension is Frobenius.
