# Abelian implies uniquely p-divisible iff pth power map is automorphism

From Groupprops

## Statement

Suppose is a prime number and is an abelian group. The following are equivalent:

- is a uniquely -divisible abelian group, i.e., for every element , there is a unique such that .
- The multiplication by map is an automorphism of , i.e., it is a bijective endomorphism.

Moreover, in this case, the *division* by map is an automorphism of , i.e., it is a bijective endomorphism. These automorphisms are inverses of each other.

## Related facts

- kth power map is bijective iff k is relatively prime to the order: In particular, if is a finite abelian group and is a prime not dividing the order of , then is uniquely -divisible and the above conclusions hold.
- Abelian implies universal power map is endomorphism (but need not be an automorphism)
- Square map is endomorphism iff abelian
- Inverse map is automorphism iff abelian
- Cube map is automorphism implies abelian