Abelian root group

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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:

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

(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:

$\langle g_1,g_2,g_3,\ldots | R_1, R_2, R_3,\ldots\rangle_P$

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.

[edit] Classification of abelian $P\;$ -root groups

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

For each $p\in P\;$ , the set $R_p=\{g\in G, \exists n\in \mathbb{N}, g^{p^n}=I\}\;$ of $p^n\;$ 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, $R_p\;$ 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 $G_U\;$ :

$G=G_U\oplus \left(R_{p_1}\oplus R_{p_2}\oplus\ldots\right)\;$

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

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

$G_U=G_T \oplus G_\infty\;$

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 $P\;$ 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, $G_U\;$ must be torsion-free, so $G_T\;$ is trivial and $G_U=G_\infty\;$ ).

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

$G_\infty=\bigoplus_{i\in I}F_{Q_i}\;$

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

$G_\infty\cong\bigoplus_{i\in I}F_P\;$

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

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

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

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:

$\langle e_1,e_2,e_3,\ldots | e_1=e_2^{p_1},e_2=e_3^{p_2},e_3=e_4^{p_3},\ldots\rangle_P$

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

[edit] Examples

The angles constructible using compass and straightedge form an abelian 2-root group under addition modulo $2\pi\;$ . 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, $\{\mathbb{Q},+\}\;$ , 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 $R_p\;$ , all of whose elements have finite order, with the usual product. This group has a presentation as an abelian $p\;$ -root group:

$\langle g \;| g \;\rangle_{\{p\}}\;$

This group is known as the Prüfer group, the p-quasicyclic group or the p^∞ group

The group $\mathbb{T}^1\;$ of complex numbers of modulus 1 forms an abelian $P\;$ -root group where $P\;$ is the set of all prime numbers. $\mathbb{T}^1\;$ may be expressed as the direct sum: