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Irreducible restriction and zeros of characters. (English) Zbl 0967.20006
It is a long known fact (by Burnside, around 1900) that an irreducible complex character $$\chi$$ of a finite group $$G$$ admits some element $$g\in G$$ satisfying $$\chi(g)=0$$ if and only if $$\chi(1)>1$$.
In this very remarkable note, the author proves the following extension to Burnside’s result. Theorem A. Suppose $$N$$ is a normal subgroup of $$G$$. Let $$\chi\in\text{Irr}(G)$$. Then $$\chi_N$$ is not irreducible if and only if $$\chi(t)=0$$ for all $$t$$ contained in some specific coset of $$N$$ in $$G$$.
As corollaries we have: B. If $$N$$ is normal in $$G$$ and $$G/N$$ a $$\pi$$-group, and if $$\chi\in\text{Irr}(G)$$ with $$\chi(s)$$ not zero on the $$\pi$$-elements $$s$$ of $$G$$, then $$\chi_N$$ is irreducible. C. If $$G=HN$$, $$H\leq G$$, $$N$$ normal in $$G$$, $$\chi\in\text{Irr}(G)$$, then $$\chi_N$$ is irreducible if $$\chi(h)\neq 0$$ whenever $$h\in H$$.
As to the proof, use has been made of so-called character-triple-isomorphisms.

##### MSC:
 20C15 Ordinary representations and characters
Full Text:
##### References:
 [1] M. Isaacs, Character Theory of Finite Groups, New York, Dover, 1994. · Zbl 0849.20004
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