Abelian root group

If G is an abelian group and P is a set of primes then G is an abelian P-root group if every element in G has a pth root for every prime p in P:

g ∈ G, p ∈ P ⇒ ∃h ∈ G, hp = g 

(with the product written multiplicatively)

If the set of primes P has only one element p, for convenience we can say G is an abelian p-root group. In a p-root group, the cardinality of the set of pth roots is the same for all elements. For any set of primes P, being a P-root group is the same as being a p-root group for every p in P.

For any specific set of primes P, the class of abelian P-root groups with abelian group homomorphisms forms a full subcategory of the category of abelian groups, but not a Serre subcategory (as the quotient of an epimorphism is an abelian group, but not necessarily an abelian P-root group). If the set of primes P is empty, the category is simply the whole category of abelian groups.

If the roots are all unique, we call G an abelian unique P-root group.

If G is an abelian unique P-root group and S is a subset of G, the abelian unique P-root subgroup generated by S is the smallest subgroup of G that contains S and is an abelian P-root group.

If G is an abelian unique P-root group generated by a set of its elements on which there are no non-trivial relations, we say G is a free abelian unique P-root group. For any particular set of primes P, two such groups are isomorphic if the cardinality of the sets of generators is the same.

An abelian P-root group can be described by an abelian P-root group presentation:

⟨g1, g2, g3, …|R1, R2, R3, …⟩P

in a similar way to those for abelian groups. However, in this case it is understood to mean a quotient of a free abelian unique P-root group rather than a free abelian group, which only coincides with the meaning for an abelian group presentation when the set P is empty.

Classification of abelian P -root groups

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

For each p ∈ P , the set Rp = {g ∈ G, ∃n ∈ ℕ, gpn = I}  of pn th roots of the identity as n  runs over all NATURAL numbers forms a subgroup of G , called the p -power torsion subgroup of G  (or more loosely the p -torsion subgroup of G ). If G  is an abelian p -root group, Rp  is also an abelian p -root group. G  may be expressed as a direct sum of these groups over the set of primes in P  and an abelian unique P -root group GU :

G = GU ⊕ (Rp1⊕Rp2⊕…) 

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

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

GU = GT ⊕ G∞ 

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 P  and a torsion-free abelian unique P -root group is an abelian unique P -root group.

In particular, if P  is the set of all prime numbers, GU  must be torsion-free, so GT  is trivial and GU = G∞ ).

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

G∞ = ⨁i ∈ IFQi 

In particular, when P  is the set of all primes,

G∞ ≅ ⨁i ∈ IFP 

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

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

∀i ∈ ℕ, pi ∉ P 

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

⟨e1, e2, e3, …|e1 = e2p1, e2 = e3p2, e3 = e4p3, …⟩P

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

Examples

  • The angles constructible using compass and straightedge form an abelian 2-root group under addition modulo 2π . Each element of this group has two 2-roots.
  • The groups of numbers with a terminating decimal expansion and addition as the product is the free abelian unique {2, 5} -root group with a single generator.
  • The group of rational numbers with addition as the product, {ℚ,  + } , is the free abelian P -root group on a single generator for P  the set of all primes.
  • For a prime p , the group of complex numbers of the form $e^{2\pi i \frac{r}{p^n}}\;$ for r  and n  natural numbers forms an abelian p -root group Rp , all of whose elements have finite order, with the usual product. This group has a presentation as an abelian p -root group:

⟨g |g ⟩{p} 

This group is known as the Prüfer group, the p-quasicyclic group or the p∞ group
  • The group 𝕋1  of complex numbers of modulus 1 forms an abelian P -root group where P  is the set of all prime numbers. 𝕋1  may be expressed as the direct sum:

𝕋1 ≅ (R2⊕R3⊕R5⊕…) ⊕ ⨁i ∈ IFP 

where each Rp  is the group defined in the previous example, FP ≅ {ℚ,  + } , and I  has the cardinality of the continuum.

See also

  • Root group