On one of Herstein’s conjectures

Abstract

Let D be a division ring with the center F. We say that N is a subgroup of D with understanding that N is in fact a subgroup of the multiplicative group D* of  D. In this note we disscus the conjecture which was posed by Herstein in 1978 [2, Conjecture 3]: If N is a subnormal subgroup of D which is radical over F, then N is contained in F. In his paper, Herstein himself  showed that the conjecture is true if N is a finite subnormal subgroup of D. However, it is not proven for the general cases. In this note, we establish some properties of subnormal subgroups in division rings which could give some information in the direction of verifying this longstanding conjecture. In particular, it is shown that  the conjecture is true for locally centrally finite division rings.