2021/09/02 by Rifkhat Zh. Aleeev, Aleeev, Rifkhat Zh., О. В. Митина +3
Computer Science · Mathematics · #20C05 (Primary) 16S34 (Secondary) #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2109.00717
openalex publication_date 2021/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is devoted to the units of integral group rings of cyclic 2-groups of small orders, namely, the orders of 2n for n<8. Immediately we should note the issues our consideration describe in the introduction in more detail. Here we will indicate the main directions of our research. Previously, we proved that the normalized group of units of an integral group ring of a cyclic 2-group of order 2n contains a subgroup of finite index, which is the direct product of the subgroup of units defined by the character with the largest character field and the subgroup of units that is isomorphic to the subgroup of units of the integer group ring of the cyclic 2-group of order 2n-1. Because of this, it is very important to study the structure of the subgroup of units defined by the character with the largest field of characters, which is the cyclotomic field Q2n obtained by adjoining a primitive 2nth root of unity to Q, the field of rational number. That subgroup of units of an integral group ring of a cyclic 2-group is isomorphic to the subgroup of the group of units of the integer ring of the specified cyclotomic field. Therefore, the research of units of an integer group ring of a cyclic 2-group is reduced to study the properties of the group of units of the integer ring of the cyclotomic field Q2n. Thus, we will study of groups of circular units of integer rings of cyclotomic fields Q2n in large part.