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Character tables and defect groups. (English) Zbl 07242353
Summary: Let \(B\) be a block of a finite group \(G\) with defect group \(D\). We prove that the exponent of the center of \(D\) is determined by the character table of \(G\). In particular, we show that \(D\) is cyclic if and only if \(B\) contains a “large” family of irreducible \(p\)-conjugate characters. More generally, for abelian \(D\) we obtain an explicit formula for the exponent of \(D\) in terms of character values. In small cases even the isomorphism type of \(D\) is determined in this situation. Moreover, it can read off from the character table whether \(| D / D^\prime | = 4\) where \(D^\prime\) denotes the commutator subgroup of \(D\). We also propose a new characterization of nilpotent blocks in terms of the character table.
Reviewer: Reviewer (Berlin)

MSC:
20C15 Ordinary representations and characters
20C20 Modular representations and characters
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EDIM; GAP
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