# Abelian root group

{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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:

${\displaystyle g\in G,p\in P\Rightarrow \exists h\in G,h^{p}=g\;}$

(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:

${\displaystyle \langle g_{1},g_{2},g_{3},\ldots |R_{1},R_{2},R_{3},\ldots \rangle _{P}}$

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 ${\displaystyle P\;}$-root groups

Suppose ${\displaystyle G\;}$ is an abelian ${\displaystyle P\;}$-root group, for some set of prime numbers ${\displaystyle P\;}$.

For each ${\displaystyle p\in P\;}$, the set ${\displaystyle R_{p}=\{g\in G,\exists n\in \mathbb {N} ,g^{p^{n}}=I\}\;}$ of ${\displaystyle p^{n}\;}$th roots of the identity as ${\displaystyle n\;}$ runs over all natural numbers forms a subgroup of ${\displaystyle G\;}$, called the ${\displaystyle p\;}$-power torsion subgroup of ${\displaystyle G\;}$ (or more loosely the ${\displaystyle p\;}$-torsion subgroup of ${\displaystyle G\;}$). If ${\displaystyle G\;}$ is an abelian ${\displaystyle p\;}$-root group, ${\displaystyle R_{p}\;}$ is also an abelian ${\displaystyle p\;}$-root group. ${\displaystyle G\;}$ may be expressed as a direct sum of these groups over the set of primes in ${\displaystyle P\;}$ and an abelian unique ${\displaystyle P\;}$-root group ${\displaystyle G_{U}\;}$:

${\displaystyle G=G_{U}\oplus \left(R_{p_{1}}\oplus R_{p_{2}}\oplus \ldots \right)\;}$

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

Each abelian unique ${\displaystyle P\;}$-root group ${\displaystyle G_{U}\;}$ is a direct sum of its torsion subgroup, ${\displaystyle G_{T}\;}$, all of which elements are of finite order coprime to all the elements of ${\displaystyle P\;}$, and a torsion-free abelian unique ${\displaystyle P\;}$-root group ${\displaystyle G_{\infty }\;}$:

${\displaystyle G_{U}=G_{T}\oplus G_{\infty }\;}$

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 ${\displaystyle P\;}$ and a torsion-free abelian unique ${\displaystyle P\;}$-root group is an abelian unique ${\displaystyle P\;}$-root group.

In particular, if ${\displaystyle P\;}$ is the set of all prime numbers, ${\displaystyle G_{U}\;}$ must be torsion-free, so ${\displaystyle G_{T}\;}$ is trivial and ${\displaystyle G_{U}=G_{\infty }\;}$).

In the case where ${\displaystyle P\;}$ includes all but finitely many primes, ${\displaystyle G_{\infty }\;}$ may be expressed as a direct sum of free abelian unique ${\displaystyle Q_{i}\;}$-root groups for a set of sets of primes ${\displaystyle Q_{i}\supseteq P\;}$.

${\displaystyle G_{\infty }=\bigoplus _{i\in I}F_{Q_{i}}\;}$

In particular, when ${\displaystyle P\;}$ is the set of all primes,

${\displaystyle G_{\infty }\cong \bigoplus _{i\in I}F_{P}\;}$

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

(This result is not true when ${\displaystyle P\;}$ has infinite complement in the set of all primes. If

${\displaystyle \forall i\in \mathbb {N} ,p_{i}\notin P\;}$

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

${\displaystyle \langle e_{1},e_{2},e_{3},\ldots |e_{1}=e_{2}^{p_{1}},e_{2}=e_{3}^{p_{2}},e_{3}=e_{4}^{p_{3}},\ldots \rangle _{P}}$

cannot be expressed as a direct sum of free abelian unique ${\displaystyle Q\;}$-root groups.)

## Examples

${\displaystyle \langle g\;|g\;\rangle _{\{p\}}\;}$
This group is known as the Prüfer group, the p-quasicyclic group or the p group
${\displaystyle \mathbb {T} ^{1}\cong \left(R_{2}\oplus R_{3}\oplus R_{5}\oplus \ldots \right)\oplus \bigoplus _{i\in I}{F_{P}}\;}$
where each ${\displaystyle R_{p}\;}$ is the group defined in the previous example, ${\displaystyle F_{P}\cong \{\mathbb {Q} ,+\}\;}$, and ${\displaystyle I\;}$ has the cardinality of the continuum.