High Quality Content by WIKIPEDIA articles! 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: (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.

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