Group with unique division

-group

A group in which the equality implies , where are any elements in the group and is any natural number. A group is an -group if and only if it is torsion-free and is such that implies for any and any natural number . An -group splits into the set-theoretic union of Abelian groups of rank 1 intersecting at the unit element. A group is an -group if and only if it is torsion-free and if its quotient group by the centre (cf. Centre of a group) is an -group. Subgroups of an -group, as well as direct and complete direct products (cf. Direct product) of -groups, are -groups. The following local theorem is valid for the class of -groups: If all finitely-generated subgroups of a group are -groups, then itself is an -group. Free groups, free solvable groups and torsion-free locally nilpotent groups (cf. Free group; Nilpotent group; Solvable group) are -groups. The class of all divisible -groups (-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)