关于有限可解群的非零元素

On Nonzero Elements of Irreducible Characters in Finite Solvable Groups

设G是一个有限可解群.若使G的所有不可约特征标都取非零值,则称G中的元素g为G的非零元素.利用非零元的生成群及置换群等方法,证明了若G是幂零群被超可解群的扩张,则这个猜想对G成立.并且将这一结果与已知的群论结果结合,证明了可解群G若有一个特征标刚好在一个共轭类上取零,则猜想成立;及一些相关结论.同时还对这个猜想的极小反例的结构进行了描述.

Let G be a finite solvable group. It is conjectured that all of nonzero elements of G lie in the fitting subgroup F（G）, the largest nilpotent normal subgroup of G. In this note, we prove that this conjecture is true for nilpotent-by-superaolvable groups. Some other related results are proved.