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