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A solvability criterion for finite groups related to character degrees
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 4, pp. 1205-1209
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Let $m>1$ be a fixed positive integer. In this paper, we consider finite groups each of whose nonlinear character degrees has exactly $m$ prime divisors. We show that such groups are solvable whenever $m>2$. Moreover, we prove that if $G$ is a non-solvable group with this property, then $m=2$ and $G$ is an extension of ${\rm A}_7$ or ${\rm S}_7$ by a solvable group.
Let $m>1$ be a fixed positive integer. In this paper, we consider finite groups each of whose nonlinear character degrees has exactly $m$ prime divisors. We show that such groups are solvable whenever $m>2$. Moreover, we prove that if $G$ is a non-solvable group with this property, then $m=2$ and $G$ is an extension of ${\rm A}_7$ or ${\rm S}_7$ by a solvable group.
DOI : 10.21136/CMJ.2020.0440-19
Classification : 20C15, 20D10
Keywords: non-solvable group; solvable group; character degree 
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Miraali, Babak; Robati, Sajjad Mahmood. A solvability criterion for finite groups related to character degrees. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 4, pp. 1205-1209. doi: 10.21136/CMJ.2020.0440-19

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