Divisible group

Definition
A divisible group is a group $$G$$ such that for any element $$g \in G$$, and any nonzero integer $$n$$, there exists $$h \in G$$ such that $$h^n = g$$.

This notion is usually discussed for abelian groups divisible abelian group, where it coincides with the notion of an injective $$\mathbb{Z}$$-module. However, the notion is useful for more general kinds of groups, particularly for nilpotent groups.

Parametric versions

 * Divisible group for a set of primes is a group where it is possible to divide any group element by any prime in the specified set of primes. For a prime set $$\pi$$, a $$\pi$$-divisible group is a group $$G$$ such that for any $$g \in G$$ and $$p \in \pi$$, there exists $$x \in G$$ such that $$x^p = g$$.

Embeddability

 * Every group is a subgroup of a divisible group