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Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Maryam KhatamiAzam Babai — 2018

Czechoslovak Mathematical Journal

Let G be a finite group. An element g G is called a vanishing element if there exists an irreducible complex character χ of G such that χ ( g ) = 0 . Denote by Vo ( G ) the set of orders of vanishing elements of G . Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo ( G ) = Vo ( M ) and | G | = | M | . Then G M . We answer in affirmative this conjecture for M = S z ( q ) , where q = 2 2 n + 1 and either q - 1 , q - 2 q + 1 or q + 2 q + 1 is a prime number, and M = F 4 ( q ) , where q = 2 n and either...

On sharp characters of type { - 1 , 0 , 2 }

Alireza AbdollahiJavad BagherianMahdi EbrahimiMaryam KhatamiZahra ShahbaziReza Sobhani — 2022

Czechoslovak Mathematical Journal

For a complex character χ of a finite group G , it is known that the product sh ( χ ) = l L ( χ ) ( χ ( 1 ) - l ) is a multiple of | G | , where L ( χ ) is the image of χ on G - { 1 } . The character χ is said to be a sharp character of type L if L = L ( χ ) and sh ( χ ) = | G | . If the principal character of G is not an irreducible constituent of χ , then the character χ 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...

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