Character degrees in blocks and defect groups. (English) Zbl 1486.20011
Let \(p\) be a prime, let \(B\) be a \(p\)-block of a finite group \(G\) with defect group \(D\), and let \(\text{cd}(B)\) denote the set of degrees of the irreducible characters in \(B\). Navarro asked whether it is always true that \(\text{dl}(D) \le |\text{cd}(B)|\) where \(\text{dl}(D)\) denotes the derived length of \(D\). The paper under review contributes to this question. It shows that the answer is positive whenever \(B\) is the principal block and \(|\text{cd}(B)| \le 3\). The proof makes use of the classification of the finite simple groups. The authors also show that \(\mathrm{dl}(D) \le |\text{ht}(B)|\) whenever \(B\) is a \(p\)-block of \(\mathfrak{S}_n\), \(\mathfrak{A}_n\) or \(\text{GL}_n(q)\) where \(p\) divides \(q\); here \(\text{ht}(B)\) denotes the set of heights of the irreducible characters in \(B\). On the other hand, the authors provide examples of finite solvable groups where the inequality \(\text{dl}(D) \le |\text{ht}(B)|\) does not hold.
Reviewer: Burkhard Külshammer (Jena)
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GAPReferences:
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