Properties
- If a divisible group is a subgroup of an abelian group then it is a direct summand.
- 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 category of -modules.
Read more about this topic: Divisible Group
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)