1
$\begingroup$

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?

Improve this question
$\endgroup$
3
  • $\begingroup$ 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. $\endgroup$
    – Tobias Kildetoft
    Commented Nov 25, 2013 at 20:08
  • $\begingroup$ 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. $\endgroup$
    – Derek Holt
    Commented Nov 26, 2013 at 8:42
  • $\begingroup$ @DerekHolt. With respect to my reference, I mean that the given group $G$ is finite and the "character" is "irreducible character". Thanks for help. $\endgroup$
    – M. Zallaghi
    Commented Nov 26, 2013 at 10:26

0

Reset to default

You must log in to answer this question.