In this interesting article the structure of the indecomposable projective modules of an indecomposable finite dimensional algebra \(\Lambda\) over an algebraically closed field K of semidihedral type with 3 non-isomorphic simple \(\Lambda\)-modules is determined. The algebra \(\Lambda\) is of semidihedral type if: (1) \(\Lambda\) is symmetric, indecomposable, not wild and has at most three simple modules; (2) the stable Auslander-Reiten quiver of \(\Lambda\) has only the following components: (i) tubes of rank at most 3, the number of 3-tubes being at most one; (ii) \({\mathbb{Z}}D_{\infty}\) and \({\mathbb{Z}}A^{\infty}_{\infty}\), and both types occur; (3) the Cartan matrix of \(\Lambda\) is non-singular. Such an algebra \(\Lambda\) has at most 3 non-isomorphic simple modules. The cases with one or two simple \(\Lambda\)-modules have been dealt with in the first part of this series of articles [ibid. 57, No.1, 109-150 (1988; Zbl 0648.20007)].
The main results of this paper show that there are 9 families of Morita equivalence classes of semidihedral type block algebras \(\Lambda\). The author describes in each case the structure of the 3 non-isomorphic indecomposable projective \(\Lambda\)-modules. Furthermore, the Cartan matrix of \(\Lambda\) is determined in each of the nine cases.
The most important application of these results is concerned with the study of 2-blocks B of a finite group G with semidihedral defect group \(\delta (B)=_ GD\). In that case, also the complete decomposition matrix of B is given. Finally it is remarked which semidihedral type block algebras \(\Lambda\) occur (up to Morita equivalence) as such blocks B. The methods of proof are based on highly original combinations of ideas from group representation theory and the theory of Auslander-Reiten graphs of finite-dimensional algebras.