Divisible group

In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups.

An abelian group ${\displaystyle (G,+)}$ is divisible if, for every positive integer ${\displaystyle n}$ and every ${\displaystyle g\in G}$, there exists ${\displaystyle y\in G}$ such that ${\displaystyle ny=g}$. An equivalent condition is: for any positive integer ${\displaystyle n}$, ${\displaystyle nG=G}$, since the existence of ${\displaystyle y}$ for every ${\displaystyle n}$ and ${\displaystyle g}$ implies that ${\displaystyle nG\supset G}$, and in the other direction ${\displaystyle nG\subset G}$ is true for every group. A third equivalent condition is that an abelian group ${\displaystyle G}$ is divisible if and only if ${\displaystyle G}$ is an injective object in the category of abelian groups; for this reason, a divisible group is sometimes called an injective group.

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