On the block algebras having only one irreducible module (Q1261759)

The author claims that a block with abelian defect group \(D\) and exactly one irreducible Brauer character is isomorphic to a complete matrix algebra over the group algebra of \(D\). There are examples, however, which show that this is not the case, in general. In fact, the construction given at the end of the paper yields, in particular, a group of order 72 with a non-principal 3-block \(B\) with inertial index 4, \(l(B) = 1\), \(k(B) = 6\) and elementary abelian defect group of order 9. Thus \(Z(B)\) has dimension 6 (not 9). Moreover, a result by \textit{T. Okuyama} and \textit{Y. Tsushima} [Osaka J. Math. 20, 33-41 (1983; Zbl 0513.20005)] shows that a block which is a complete matrix algebra over its center has abelian defect groups and inertial index 1.