Open AccessON ONE OF HERSTEIN'S CONJECTURES
Author(s) -
Thin Van Nguyen,
Hai Xuan Bui
Publication year - 2009
Publication title -
khoa học công nghệ
Language(s) - English
Resource type - Journals
ISSN - 1859-0128
DOI - 10.32508/stdj.v12i11.2307
Subject(s) - conjecture , division ring , mathematics , combinatorics , center (category theory) , division (mathematics) , multiplicative group , multiplicative function , ring (chemistry) , maximal subgroup , group (periodic table) , normal subgroup , arithmetic , physics , mathematical analysis , crystallography , chemistry , organic chemistry , quantum mechanics
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.