On the Polarization Levels of Automorphic-Symmetric Channels

It is known that if an Abelian group operation is used in an Arikan-style construction, multilevel polarization occurs. An open problem in polarization theory is to determine the polarization levels of a given channel. In this paper, we discuss the polarization levels of a family of channels that we call automorphic-symmetric channels. We show that the polarization levels of an automorphic-symmetric channel are determined by characteristic subgroups. In particular, if the group that is used does not contain any non-trivial characteristic subgroup, we only have two-level polarization to almost perfect and almost useless channels.