For a character $\chi $ of a finite group G, the number $\chi ^c(1)={[G:{\textrm {ker}}\chi ]}/{\chi (1)}$ is called the co-degree of $\chi $ . A finite group G is an ${\textrm {NDAC}} $ -group (no divisibility among co-degrees) when $\chi ^c(1) \nmid \phi ^c(1)$ for all irreducible characters $\chi $ and $\phi $ of G with $1< \chi ^c(1) < \phi ^c(1)$ . We study finite groups admitting an irreducible character whose co-degree is a given prime p and finite nonsolvable ${\textrm {NDAC}} $ -groups. Then we show that the finite simple groups $^2B_2(2^{2f+1})$ , where $f\geq 1$ , $\mbox {PSL}_3(4)$ , ${\textrm {Alt}}_7$ and $J_1$ are determined uniquely by the set of their irreducible character co-degrees.

中文翻译：

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不可约字符共度之间的不可分割性

对于一个角色$\吃$有限群的G， 号码$\chi ^c(1)={[G:{\textrm {ker}}\chi ]}/{\chi (1)}$称为共度$\吃$. 有限群G是一个${\textrm {NDAC}} $-group（在 co-degrees 之间不可分割），当$\chi ^c(1) \nmid \phi ^c(1)$对于所有不可约字符$\吃$和$\phi $的G和$1< \chi ^c(1) < \phi ^c(1)$. 我们研究承认一个不可约特征的有限群，其共度是给定的素数p和有限不可解${\textrm {NDAC}} $-团体。然后我们证明有限单群$^2B_2(2^{2f+1})$， 在哪里$f\geq 1$,$\mbox {PSL}_3(4)$,${\textrm {Alt}}_7$和$J_1$由它们的不可约特征共度的集合唯一确定。