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?)

$x^y$ is not even well-defined for many real arguments.  (For example, $(-2)^\frac12$.) You would have to begin by restricting the domain to non-negative reals.
But even then, exponentiation is not in general associative, since $x^{(y^z)} \ne  (x^y)^z$, and to make it associative you would have to restrict the domain so severely that what is left would not be very interesting.
