Abelian-extensible automorphism not implies power map

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Statement

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

Facts used

  1. Injective module implies every automorphism is infinity-extensible

Proof

For both these proofs, we use the fact that the group \mathbb{Q} of rational numbers is a divisible group, and hence, is injective as a \mathbb{Z}-module. Since a direct sum of injective modules is injective, the group \mathbb{Q} \oplus \mathbb{Q} 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 \mathbb{Q} \oplus \mathbb{Q}, consider the automorphism (x,y) \mapsto (y,x), 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.