Divisible group

A group in which for any element $g$ and for any integer $n\neq0$ the equation $x^n=g$ is solvable. The group is usually understood to be Abelian. Important examples of divisible groups are the additive group of all rational numbers and the group of all complex roots of unity of degrees $p^k$, $k=1,2,\ldots,$ where $p$ is a prime number (the quasi-cyclic group). Any Abelian divisible group splits into a direct sum of groups each of which is isomorphic to one of the groups mentioned in the examples. Much less is known about non-Abelian divisible groups (also called complete groups). Any divisible group, except the identity group, is infinite. Any group is imbeddable in a suitable divisible group. If the equations stated in the definition of a divisible group have a unique solution, the group is called a $D$-group. Examples, in particular, are locally nilpotent divisible torsion-free groups.

References

[1]	A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2]	M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)

Comments

An Abelian group is divisible if and only if, regarded as a $\mathbf Z$-module, it is injective (cf. Injective module). Let $\mathbf Q_p$ be the field of $p$-adic numbers and $\mathbf Z_p$ its ring of integers. Then the quasi-cyclic group for the prime $p$ is the quotient group $\mathbf Q_p/\mathbf Z_p$ which is also the injective limit $\lim_\to\mathbf Z/(p^n)$ for the imbeddings $\mathbf Z/(p^k)\to\mathbf Z/(p^{k+l})$, $1\mapsto p^l$.

See also Split group; Splittable group.

References

[a1]	L. Fuchs, "Infinite abelian groups" , 1 , Acad. Press (1970) pp. Chapt. 4