If $G$ is a finite group and $T$ is an automorphism of it, such that:
$T(x)=x \iff x=e$
Prove that:
(i) $G=\{x^{-1}T(x):x\in G\}$
(ii) If $T^2=id$ then $G$ is abelian.
I know that this question DOES have answer, at least i can provide these links:
Prove that every element of a group G can be represented as $g = x^{-1}(xT)$ for some x $\in$ G?
$xT=x$ iff $x=e$ implies that every $g$ may be written as $x^{-1}(xT)$ for some $x\in G$.
But the problem is the notation. I can't understand it. My notation is different. I'm confused at reading that answers. Because they both use some sort of mysterious notation.
Can someone please clarify the answer?