  A finite group G with center Z is of central type if there exists a fully
ramified character $\lambda\in\mathrm{Irr}(Z)$, 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-Sp\"ath-Tiep under the assumption
that $p\ne 5$. 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.
