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Let be a finite group. An element is called a vanishing element if there exists an irreducible complex character of such that . Denote by the set of orders of vanishing elements of . Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let be a finite group and a finite nonabelian simple group such that and . Then . We answer in affirmative this conjecture for , where and either , or is a prime number, and , where and either...
For a complex character of a finite group , it is known that the product is a multiple of , where is the image of on . The character is said to be a sharp character of type if and . If the principal character of is not an irreducible constituent of , then the character is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups with normalized sharp characters of type . Here we prove that such a group with nontrivial center is...
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