Structure Theorem of Divisible Groups
Let G be a divisible group. One can easily see that 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. So
As a quotient of a divisible group, G/Tor(G) is divisible. Moreover, it is torsion-free. Thus, it is a vector space over Q and so there exists a set I such that
The structure of the torsion subgroup is harder to determine, but one can show that for all prime numbers p there exists such that
where is the p-primary component of Tor(G).
Thus, if P is the set of prime numbers,
Read more about this topic: Divisible Group
Famous quotes containing the words structure, theorem, divisible and/or groups:
“The structure was designed by an old sea captain who believed that the world would end in a flood. He built a home in the traditional shape of the Ark, inverted, with the roof forming the hull of the proposed vessel. The builder expected that the deluge would cause the house to topple and then reverse itself, floating away on its roof until it should land on some new Ararat.”
—For the State of New Jersey, U.S. public relief program (1935-1943)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“Knowledge, like matter, [my father] would affirm, was divisible in infinitum;Mthat the grains and scruples were as much a part of it, as the gravitation of the whole world.—In a word, he would say, error was error,—no matter where it fell,—whether in a fraction,—or a pound,—’twas alike fatal to truth.”
—Laurence Sterne (1713–1768)
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)