Blocks of tame representation type and related algebras / Karin Erdmann

The study of blocks with cyclic defect group was carried out by R. Brauer, E. Dade and G. Janusz and has become a standard ingredient in the modular representation theory of finite groups. Ideally there would be, for each p-group P, a complete account available of p-blocks (of arbitrary groups) with defect group isomorphic to P. This 300-page book assembles work by various people, notably the author, that gives an almost complete account of the 2-blocks (of arbitrary groups) with dihedral, semidihedral or generalized quaternion defect groups.
The work is sufficiently complete to enable a new representation-theoretic proof of the following theorem (Brauer-Suzuki, 1959) to emerge: A finite group whose Sylow subgroup is generalised quaternion is not simple. On the other hand, the methods used are anything but traditional group representation theory. The concepts of Auslander-Reiten sequences, the stable Auslander-Reiten quivers and tame representation type come in, together with the convenient (and classical) fact that such blocks have at most three simple modules.
There are two principal unresolved gaps left in the subject. They are the "socle constants'', which constitute a minor ambiguity in the Morita equivalence classes of the block algebras, and explicit parametrization of the indecomposable modules in the generalized quaternion defect group case.
Indeed it may be said that the book will be of greater interest to those seeking knowledge about representing algebras than to the finite group theorists. The group theorists will note that the tame representation type hypothesis is used so heavily that no extension to any other class of defect groups is in sight. The algebra representers will find the book a useful guide to the current work in the subject. (MathSciNet)