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Finite group elements where no irreducible character vanishes. (English) Zbl 0959.20009
Let \(G\) be a finite group and let \(\chi\in\text{Irr}(G)\) be an irreducible complex character of \(G\). If \(\chi(g)\neq 0\) for all \(g\in G\), then \(\chi\) has to be a linear character by a well-known result of Burnside. In the paper under review the authors investigate the dual question, i.e., what can be said about \(g\in G\) for which \(\chi(g)\neq 0\) for all \(\chi\in\text{Irr}(G)\). Such elements are called nonvanishing. First, note that it may happen that the trivial element is the only nonvanishing element; for instance this is the case for groups of Lie type since they have \(p\)-blocks of defect zero for all primes \(p\). On the other hand, if \(g\in Z(G)\), then obviously \(g\) is nonvanishing, but the converse does not hold true. It is even shown in the article that for each prime \(p\) there exists a solvable group containing a nonvanishing \(p\)-element that does not lie in any Abelian normal subgroup. Besides many interesting results the main theorem is as follows: Suppose that \(G\) is solvable and \(x\in G\) nonvanishing. Then, if \(F(G)\) denotes the Fitting group of \(G\), \(xF(G)\in G/F(G)\) has 2-power order. In particular, if \(x\) is of odd order, then \(x\in F(G)\). Moreover, if \(G\) is not nilpotent, then \(x\) lies in the penultimate term of the ascending Fitting series.

MSC:
20C15 Ordinary representations and characters
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D25 Special subgroups (Frattini, Fitting, etc.)
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