Extension need not preserve powering
Statement
It is possible to have a group , a normal subgroup
, and a prime number
such that the following hold:
-
is powered over
.
- The quotient group
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.