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
Famous quotes containing the word definition:
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)