Rationally powered not implies nilpotent

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This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., rationally powered group) need not satisfy the second group property (i.e., nilpotent group)
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It is possible to construct a rationally powered group (i.e., a group that is uniquely p-divisible for all primes p) that is not a nilpotent group.


Further information: GAPlus(1,R)

The easiest example is the following group G = GA^+(1,\R): G is the group of all (affine) linear maps from \R to \R with positive leading coefficient, under composition. Explicitly:

G = \{ x \mapsto ax + b \mid a,b \in \R, a > 0 \}

Then, we have that: