Critical exponent and critical blow-up for quasilinear parabolic equations

B ₂-groups are special (torsion-free) abelian Butler groups. The interest in this class of groups comes from representation theory. A particular functor, also called Butler functor, connects algebraic properties of the category of free abelian groups with (a few) distinguished subgroups with these Butler groups. This helps to understand Butler groups and caused lots of activities on Butler groups. Butler groups were originally defined for finite rank, however a homological connection discovered by Bican and Salce opened the investigation of Butler groups of infinite rank. Despite the fact that classifications of Butler groups are possible under restriction even for infinite rank (see a forthcoming paper by Files and Göbel [Mathematische Zeitschrift]), general structure theorems are impossible. This is supported by the following very special case of the Main Theorem of this paper, showing that any ring with a free additive group is an endomorphism ring of a Butler group. The result implies the existence of large indecomposable or of large superdecomposable Butler groups as well as the existence of counter-examples for Kaplansky’s test problems.