Is $G = (\{x \in \mathbb{R} | x > 0\}, +)$ with $a + b \to a^b$ a group? Hi MathOverflow community,
I just stumbled upon the question in the title while learning for my Algebra exam. Intuitively, I would say no since:
$$
e+a = e^a = a \\
e = \sqrt[a]{a}
$$
which depends on a and is therefore not a single neutral element for all elements of the set. Is that the right argument or is there a way to argue that more elegantly?

Another argument can be that it does not hold the associativity property, because $(a^b)^c \neq a^{(b^c)}$.
