Abelian Root Group - Classification of Abelian -root Groups

Classification of Abelian -root Groups

Suppose is an abelian -root group, for some set of prime numbers .

For each, the set of th roots of the identity as runs over all natural numbers forms a subgroup of, called the -power torsion subgroup of (or more loosely the -torsion subgroup of ). If is an abelian -root group, is also an abelian -root group. may be expressed as a direct sum of these groups over the set of primes in and an abelian unique -root group :

Conversely any abelian group that is a direct sum of an abelian unique -root group and a direct sum over of abelian -root groups all of whose elements have finite order is an abelian -root group.

Each abelian unique -root group is a direct sum of its torsion subgroup, all of which elements are of finite order coprime to all the elements of, and a torsion-free abelian unique -root group :

G_∞ is simply the quotient of the group G by its torsion subgroup.

Conversely any direct sum of a group all of whose elements are of finite order coprime to all the elements of and a torsion-free abelian unique -root group is an abelian unique -root group.

In particular, if is the set of all prime numbers, must be torsion-free, so is trivial and ).

In the case where includes all but finitely many primes, may be expressed as a direct sum of free abelian unique -root groups for a set of sets of primes .

In particular, when is the set of all primes,

a sum of copies of the rational numbers with addition as the product.

(This result is not true when has infinite complement in the set of all primes. If

is an infinite set of primes in the complement of then the abelian unique -root group which is the quotient by its torsion subgroup of the group with the following presentation:

cannot be expressed as a direct sum of free abelian unique -root groups.)