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On the semiproportional character conjecture. (Russian, English) Zbl 1096.20009

Sib. Mat. Zh. 46, No. 2, 299-314 (2005); translation in Sib. Math. J. 46, No. 2, 233-245 (2005).
Characters \(\varphi\) and \(\psi\) of a finite group \(G\) are called semiproportional if they are not proportional and there is a normal subset \(M\) of \(G\) such that \(\varphi|_M\) is proportional to \(\psi|_M\) and \(\varphi|_{G\setminus M}\) is proportional to \(\psi|_{G\setminus M}\).
The author establishes some properties of a finite group \(G\) having a pair of semiproportional irreducible characters \(\varphi\) and \(\psi\) and, in particular, obtains some results on the order of \(G\) and on the kernels of \(\varphi\) and \(\psi\). He also considers the conjecture that semiproportional irreducible characters of a finite group must be of the same degree. He proves that the conjecture is true for the direct product of two groups if it is true for each of the factors. He verifies the conjecture in the case when \(G\) is the direct product of a 2-group and a group of odd order.

MSC:

20C15 Ordinary representations and characters
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