NONDIVISIBILITY AMONG IRREDUCIBLE CHARACTER CO-DEGREES

Abstract

AbstractFor 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.