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Keywords:
sharp character; sharp pair; finite group
sharp character; sharp pair; finite group
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.
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