Article
| Title: | On sharp characters of type $\{ -1,0,2 \}$ (English) |
| Author: | Abdollahi, Alireza |
| Author: | Bagherian, Javad |
| Author: | Ebrahimi, Mahdi |
| Author: | Khatami, Maryam |
| Author: | Shahbazi, Zahra |
| Author: | Sobhani, Reza |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 1081-1087 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | For a complex character $ \chi $ of a finite group $ G $, it is known that the product $ {\rm sh}(\chi ) = \prod _{ l \in L(\chi )} (\chi (1) - l) $ is a multiple of $ |G| $, where $ L(\chi ) $ is the image of $ \chi $ on $ G-\{1\}$. The character $ \chi $ is said to be a sharp character of type $ L $ if $ L=L(\chi ) $ and $ {\rm sh} (\chi )=|G| $. If the principal character of $G$ is not an irreducible constituent of $\chi $, then the character $\chi $ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups $G$ with normalized sharp characters of type $\{-1,0,2\}$. Here we prove that such a group with nontrivial center is isomorphic to the dihedral group of order 12. (English) |
| Keyword: | sharp character |
| Keyword: | sharp pair |
| Keyword: | finite group |
| MSC: | 20C15 |
| idZBL: | Zbl 07655784 |
| idMR: | MR4517597 |
| DOI: | 10.21136/CMJ.2022.0356-21 |
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| Date available: | 2022-11-28T11:38:05Z |
| Last updated: | 2023-04-11 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151131 |
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| Reference: | [9] Nozawa, S.: Sharp character of finite groups having prescribed values.Tsukuba J. Math. 16 (1992), 269-277. Zbl 0819.20009, MR 1178680, 10.21099/tkbjm/1496161845 |
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| Reference: | [11] Yoguchi, T.: On determining the sharp characters of finite groups.JP J. Algebra Number Theory Appl. 11 (2008), 85-98. Zbl 1169.20006, MR 2458670 |
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