Definition
An abelian group G is divisible if and only if, for every positive integer n and every g in G, there exists y in G such that ny = g. An equivalent condition is: for any positive integer n, nG = G, since the existence of y for every n and g implies that nG ⊇ G, and in the other direction nG ⊆ G is true for every group. A third equivalent condition is that an abelian group G is divisible if and only if G is an injective object in the category of abelian groups; for this reason, a divisible group is sometimes called an injective group.
An abelian group is p-divisible for a prime p if for every positive integer n and every g in G, there exists y in G such that pny = g. Equivalently, an abelian group is p-divisible if and only if pG = G.
Read more about this topic: Divisible Group
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