Geodesic
On sharp characters of type $\{ -1,0,2 \}$
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 1081-1087
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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.
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.
DOI : 10.21136/CMJ.2022.0356-21
Classification : 20C15
Keywords: sharp character; sharp pair; finite group 
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     title = {On sharp characters of type $\{ -1,0,2 \}$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1081--1087},
     year = {2022},
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Abdollahi, Alireza; Bagherian, Javad; Ebrahimi, Mahdi; Khatami, Maryam; Shahbazi, Zahra; Sobhani, Reza. On sharp characters of type $\{ -1,0,2 \}$. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 1081-1087. doi: 10.21136/CMJ.2022.0356-21

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