# Abelian-extensible automorphism not implies power map

## Contents

## Statement

We can have an abelian group and an abelian-extensible automorphism of such that is *not* a power map.

## Facts used

## Proof

For both these proofs, we use the fact that the group of rational numbers is a divisible group, and hence, is injective as a -module. Since a direct sum of injective modules is injective, the group is also injective. We combine these ideas with fact (1).

### Rational multiplication on the rational numbers

On the group of rational numbers, consider an automorphism give by multiplication by a non-integer rational number. Such an automorphism is extensible (by fact (1), all automorphisms are extensible), but it is not a power map.

### Coordinate exchange on two copies of the rational numbers

On the group , consider the automorphism , the coordinate exchange automorphism. Such an automorphism is extensible (by fact (1), all automorphisms are extensible), but it is not a power map. In fact, it is not even in the group of automorphisms *generated* by power maps.