On the largest character codegree of a finite group.

Let G be a finite group and Irr (G) be the set of irreducible characters of G. The codegree of an irreducible character χ of the group G is defined as cod (χ) = | G : ker (χ) | / χ (1) . Let b c (G) be the largest codegree of G. In this paper, we study how the structure of a group G is bounded by its largest codegree b c (G) . Firstly, we give a criterion for solvability: If b c (G) < 20 , then G is solvable. Secondly, we consider the nonsolvable groups G with a small b c (G) and prove that, if b c (G) < 56 , then G is isomorphic to A 5 , S 5 or A 5 × A where A is an elementary abelian 2-group. [ABSTRACT FROM AUTHOR]