# Semidirect product need not preserve powering

## Statement

It is possible to have a group , a complemented normal subgroup (i.e., is an internal semidirect product involving ), and a prime number such that the following hold:

- is powered over .
- The quotient group (which is also isomorphic to any permutable complement to in ) is also powered over .
- is not powered over .

Note that it is not possible to construct finite examples, because in the finite case, being powered over a prime is equivalent to not dividing the order (see kth power map is bijective iff k is relatively prime to the order).

## Proof

Below is an example where both and are rationally powered (i.e., powered over all primes), but is not powered over any prime. There may be simpler examples.

Let be the subgroup inside . Recall that the group GAPLus(1,R) is the group of affine maps from to where the multiplication is positive, i.e.:

Let be the vector space over generated by as a subset of . Note that is a subring of , because its generating set is a multiplicative monoid:

Explicitly, it is the set of maps:

Let be the base of the semidirect product here, so it is isomorphic to .

- is powered over all primes: That's because it is isomorphic to , a vector space over .
- is powered over all primes: That is because it is isomorphic to , which is isomorphic to .
- is not powered over any prime. Consider the element of of the form (here, is transcendental). For a prime , the unique root of this in is:

We would like to claim that the number is not an element of , so that this root is not in . This can be deduced from the fact that is trancendental.