Let $\sqrt[3]{a}$ indicate the real cube root of a number $a \in \Bbb{R}$.
Then the set of all cube roots forms a group under $\sqrt[3]{a} \oplus \sqrt[3]{b} = \sqrt[3]{a^3 + b^3}$. This group is not isomorphic to $(\Bbb{R}, +)$ I don't think.
What is the significance of this group?
This operation is not a group operation, since it is not associative. For instance, $$1\oplus (1\oplus -1)=1\oplus0=1$$ is different from $$(1\oplus 1)\oplus -1=\sqrt[3]{2}\oplus -1=\sqrt[3]{7}.$$
This operation is much easier to think about if you ignore the cube roots and write it as $a\oplus b=a^3+b^3$ (which gives an isomorphic structure, since taking cube roots is a bijection). This makes it more obvious why you should not expect it to be associative, or very nice algebraically at all.