Group with unique division

$R$-group

A group in which the equality $x^n=y^n$ implies $x=y$, where $x,y$ are any elements in the group and $n$ is any natural number. A group $G$ is an $R$-group if and only if it is torsion-free and is such that $x^ny=yx^n$ implies $xy=yx$ for any $x,y\in G$ and any natural number $n$. An $R$-group splits into the set-theoretic union of Abelian groups of rank 1 intersecting at the unit element. A group is an $R$-group if and only if it is torsion-free and if its quotient group by the centre (cf. Centre of a group) is an $R$-group. Subgroups of an $R$-group, as well as direct and complete direct products (cf. Direct product) of $R$-groups, are $R$-groups. The following local theorem is valid for the class of $R$-groups: If all finitely-generated subgroups of a group $G$ are $R$-groups, then $G$ itself is an $R$-group. Free groups, free solvable groups and torsion-free locally nilpotent groups (cf. Free group; Nilpotent group; Solvable group) are $R$-groups. The class of all divisible $R$-groups ($D$-groups, cf. also Divisible group) forms a variety of algebras under the operations of multiplication and division.

References

[1]	A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)