Groups of p-central type

2023/05/31 by Sambale, Benjamin
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2305.19955

Abstract

A finite group G with center Z is of central type if there exists a fully ramified character λ\inIrr(Z), i.e. the induced character 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≠ 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.

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