Define any non-commutative operation for the group $\left({\mathbb{R}, \circ}\right)$

Question

Let $\mathbb{R}$ denote the set of real numbers. Given that $\left({\mathbb{R}, \circ}\right)$ is a group, provide any definition for $\circ$, so that $\circ$ is not commutative.

Answer 1

Take $f:\mathbb R \rightarrow GL(\mathbb R,2)$ any bijection. Remember $GL(\mathbb R,2)$ is the group of invertible $2\times 2 $ matrices.

We now define $x\star y=f^{-1}(f(x) f(y))$ (where $f(x)$ and $f(y)$ are multipled like normal matrices).

Then cleary $(\mathbb R,\star)$ is isomorphic to $GL(\mathbb R,2)$

as $f(x\star y) = f(f^{-1}(f(x)f(y))=f(x)f(y)$.

Since $GL(\mathbb R,2)$ is not abelian we have $(\mathbb R, \star)$ is not abelian.

In general, if the set $X$ has the same cardinality as group $G$, we can copy the structure of $G$ to the set $X$.