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According to

Lev Kazarin, On Thompson’s Theorem, Journal of Algebra 220, 574–590 (1999)

we know that:

[Corollary 5.3]:Let $$cd(G)=\{\chi(1)|\chi\in Irr(G)\}=\{1,f_1,\dots,f_n,d\}, \;\;n\gt0,$$ where $$f_1\lt \dots\lt f_n,\;\;\; f_1|f_2|\dots|f_n,\;\;\;(d,f_n)=1.$$ If $Irr(G)$ contains only one character of degree $d$, then the following assertions hold:

(a) $G$ contains a normal subgroup $N$ such that $G/N$ is cyclic of order $d$.

(b) $cd(N)=\{1,f_1,\dots,f_n\}$ so that $N$ has an ordered Sylow tower.

Do you know any clear classification of groups in which there would exist a unique non-linear character of a given degree?

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I am not sure I know what you mean. This does not seem to be an example of the existence of a non-linear character of a unique degree, but rather the existence of a unique character of a given degree. –  Tobias Kildetoft Nov 25 '13 at 20:08
    
thanks a lot @TobiasKildetoft, I corrected it. –  M. Zallaghi Nov 25 '13 at 21:00
    
I am still unsure exactly what the question is. You would do better to write the question using formal mathematical language rather than trying to express it in words. You are looking for a classification of (presumably finite?) groups for which there exists an integer $d>0$ such that there is a unique (presumably irreducible?) character of degree $d$? I would be very surprised if it was possible to classify all finite groups satisfying that condition. –  Derek Holt Nov 26 '13 at 8:42
    
@DerekHolt. With respect to my reference, I mean that the given group $G$ is finite and the "character" is "irreducible character". Thanks for help. –  M. Zallaghi Nov 26 '13 at 10:26
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