Editing Divisible group (section)

==Definition==

An abelian group <math>(G, +)</math> is '''divisible''' if, for every positive integer <math>n</math> and every <math>g \in G</math>, there exists <math>y \in G</math> such that <math>ny=g</math>.<ref>Griffith, p.6</ref>  An equivalent condition is:  for any  positive integer <math>n</math>, <math>nG=G</math>, since the existence of <math>y</math> for every <math>n</math> and <math>g</math> implies that <math>n G\supseteq G</math>, and the other direction <math>n G\subseteq G</math> is true for every group. A third equivalent condition is that an abelian group <math>G</math> is divisible if and only if <math>G</math> 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 <math>p</math>-'''divisible''' for a [[prime number|prime]] <math>p</math> if for every <math>g \in G</math>, there exists <math>y \in G</math> such that <math>py=g</math>.  Equivalently, an abelian group is <math>p</math>-divisible if and only if <math>pG=G</math>.