Units of integral group rings of cyclic $2$-groups

TL;DR

This paper investigates the structure of units in integral group rings of small cyclic 2-groups, focusing on the subgroup related to the cyclotomic field generated by roots of unity, to better understand their algebraic properties.

Contribution

It provides a detailed analysis of the units in integral group rings of cyclic 2-groups of order less than 256, emphasizing the subgroup associated with the cyclotomic field Q_{2^n}.

Findings

Identified the structure of units related to the cyclotomic field Q_{2^n}.

Reduced the problem to studying units in cyclotomic integer rings.

Provided insights into the subgroup of units of finite index.

Abstract

This paper is devoted to the units of integral group rings of cyclic $2$-groups of small orders, namely, the orders of $2^n$ 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 $2^n$ 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 $2^{n-1}$. 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 $Q_{2^n}$ obtained by adjoining a primitive $2^n$th 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 $Q_{2^n}$. Thus, we will study of groups of circular units of integer rings of cyclotomic fields $Q_{2^n}$ in large part.