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 is divisible if, for every positive integer and every , there exists such that . An equivalent condition is: for any positive integer , , since the existence of for every and implies that , and the other direction is true for every group. A third equivalent condition is that an abelian group is divisible if and only if is an injective object in the ; for this reason, a divisible group is sometimes called an injective group. An abelian group is -divisible for a prime if for every , there exists such that . Equivalently, an abelian group is -divisible if and only if . The rational numbers form a divisible group under addition. More generally, the underlying additive group of any vector space over is divisible. Every quotient of a divisible group is divisible. Thus, is divisible. The p-primary component of , which is isomorphic to the p-quasicyclic group , is divisible. The multiplicative group of the complex numbers is divisible. Every existentially closed abelian group (in the model theoretic sense) is divisible. If a divisible group is a subgroup of an abelian group then it is a direct summand of that abelian group. Every abelian group can be embedded in a divisible group. Non-trivial divisible groups are not finitely generated. Further, every abelian group can be embedded in a divisible group as an essential subgroup in a unique way. An abelian group is divisible if and only if it is p-divisible for every prime p. Let be a ring. If is a divisible group, then is injective in the of -modules. Let G be a divisible group. Then the torsion subgroup Tor(G) of G is divisible. Since a divisible group is an injective module, Tor(G) is a direct summand of G.