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:
“One theme links together these new proposals for family policy—the idea that the family is exceedingly durable. Changes in structure and function and individual roles are not to be confused with the collapse of the family. Families remain more important in the lives of children than other institutions. Family ties are stronger and more vital than many of us imagine in the perennial atmosphere of crisis surrounding the subject.”
—Joseph Featherstone (20th century)
“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)
“We operate exclusively with things that do not exist, with lines, surfaces, bodies, atoms, divisible time spans, divisible spaces—how could explanations be possible at all when we initially turn everything into images, into our images!”
—Friedrich Nietzsche (1844–1900)
“Instead of seeing society as a collection of clearly defined “interest groups,” society must be reconceptualized as a complex network of groups of interacting individuals whose membership and communication patterns are seldom confined to one such group alone.”
—Diana Crane (b. 1933)