Groups of p-central type

A finite group G with center Z is of central type if there exists a fully ramified character $\lambda \in \mathrm{I}\mathrm{r}\mathrm{r}(Z)$, i.e. the induced character ${\lambda }^G$ is a multiple of an irreducible character. Howlett-Isaacs have shown that G is solvable in this situation. A corresponding theorem for p-Brauer characters was proved by Navarro-Späth-Tiep under the assumption that $p\mathrm{\ne }5$. We show that there are no exceptions for p=5, i.e. every group of p-central type is solvable. Gagola proved that every solvable group can be embedded in G/Z for some group G of central type. We generalize this to groups of p-central type. As an application we construct some interesting non-nilpotent blocks with a unique Brauer character. This is related to a question by Kessar and Linckelmann.