A variation on a theorem of Gluck
Gluck proved that any finite group G has an abelian subgroup A such that |G : A| is bounded by a polynomial function of the largest degree of the complex irreducible characters of G. This improved on a previous bound of Isaacs and Passman. Moretó (J. Algebra 301:274–279, 2006) presented a variation of this result that looks at the number of prime factors and obtained an almost quadratic bound. In this note, we improve the result of Moretó to almost linear.
KeywordsCharacter degrees Prime divisors Arithmetic properties
Mathematics Subject Classification20C15
The project is supported by a Texas State University Research Enhancement Program. I am grateful to the referee for his/her valuable suggestions which improved the manuscript.