Exponential group? Considering a suitable set of numbers, you can construct a group using addition, and you can construct another group using multiplication. My question: Can you construct a group using exponentiation?


*

*$(\mathbb{R}, +)$ is a group.

*$(\mathbb{R} \backslash 0, \times)$ is another group.

*Does there exist some $S \subseteq \mathbb{R}$ such that $(S, \uparrow)$ is a group?
(Here $x \uparrow y = x^y$.)
I'm going to guess "no", since exponentiation is asymmetric and hence needs two inverses (roots and logarithms). So maybe you can have some other group-like structure? (But which one?)

This isn't quite what you want, but it's a nice example and deserves to be widely known: the reals exceeding 1 form a (commutative!) group under the operation given by $a*b=a^{\log b}$. 
