Irreducible restriction and zeros of characters.

*(English)*Zbl 0967.20006It 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.

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.

Reviewer: R.W.van der Waall (Amsterdam)

##### MSC:

20C15 | Ordinary representations and characters |

##### Keywords:

representations; zeros of characters; irreducible complex characters; finite groups; \(\pi\)-groups; \(\pi\)-elements; character-triple-isomorphisms
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\textit{G. Navarro}, Proc. Am. Math. Soc. 129, No. 6, 1643--1645 (2001; Zbl 0967.20006)

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##### References:

[1] | M. Isaacs, Character Theory of Finite Groups, New York, Dover, 1994. · Zbl 0849.20004 |

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