Groups of p-central type

Abstract

AbstractA finite group G with center Z is of central type if there exists a fully ramified character $$\lambda \in \textrm{Irr}(Z)$$ λ ∈ Irr ( Z ) , i. e. the induced character $$\lambda ^G$$ λ 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 $$p\ne 5$$ p ≠ 5 . We show that there are no exceptions for $$p=5$$ 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.