$\mathbb Q$-completions of free solvable groups

Ch. K. Gupta ; N. S. Romanovskii

Algebra i logika, Tome 54 (2015) no. 2, pp. 193-211

Résumé

A group $G$ is said to be complete if, for any natural $n$ and any element $g\in G$, an equation $x^n=g$ is solvable in $G$. If every such equation in the group has at most one solution, then we say that the condition for uniqueness of root extraction is satisfied. A complete group with unique root extraction can be treated as a $\mathbb Q$-power group since it admits an operation of raising an element to any rational power. Let a group $G$ be embedded in a complete group $H$ with unique root extraction, and let $H$ be generated as a $\mathbb Q$-group by the set $G$. Then $H$ is called a $\mathbb Q$-completion of $G$. We prove that every $m$-rigid group $G$ is independently embedded in a complete $m$-rigid group. Under the specified condition for independence of an embedding, the $\mathbb Q$-completion of the group $G$ in the class of rigid groups is defined uniquely up to $G$-isomorphism. It is stated that the centralizer of any element of an independent $\mathbb Q$-completion of a free solvable group which does not belong to the last nontrivial member of a rigid series of this completion is isomorphic to the additive group of a field $\mathbb Q$ of rational numbers.

Détail

Mots-clés : $m$-rigid group
Keywords: free solvable group, $\mathbb Q$-completion.