Giving an example of an automorphism for a group

Question

Consider a group $S:(\Bbb R\neq 0, \phi)$ consisting of all real numbers except zero, and operation $\phi(v,w)=vw$. Find an automorphism of $S$.

My process: An automorphism $\gamma$ is a bijection such that $\phi(\gamma(v),\gamma(w))=\phi(v,w)$.

Let $\gamma(a)=-a$. So $\phi(\gamma(v),\gamma(w))=\gamma(v)\gamma(w)=(-v)(-w)=vw=\phi(v,w)$

Thus, $\gamma$ is an automorphism of $\phi$. Is this correct? I do not have too much experience in Group Theory so I can't really tell.

Answer 1

Your example is not an automorphism.

For example: (Using $\cdot$ as an infix symbol for $\phi$)

$\gamma(2\cdot 3)=\gamma(6)=-6$; but $\gamma(2)\cdot \gamma(3)=(-2)\cdot (-3)=6$.

Note: An automorphism of a group must send the group identity element ($1$ in this example) to itself.

Answer 2

Your definition of automorphism is wrong: it should be $$\phi(\gamma(v),\gamma(w))=\gamma(\phi(v, w)).$$ So in fact the map you've described isn't an automorphism. (The map $x\mapsto -x$ is an automorphism of $(\mathbb{R}, +)$, but that's a different group.)