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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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##### References:
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