Let T be an automorphism of finite group G & order of T is 2. Taking any element in G say x then $$xT = xT^{-1} = x^{-1}T$$ which implies $x=x^{-1}$ by taking $xTT^{-1} = x^{-1}TT^{-1}$. But since not every automorphism of order 2 results $x=x^{-1}$ for all x in G (Particularly, If I consider a finite Group G with at the least one element is of order greater than 2 and defining T as $xT=x^{-1}$), The above implication is contradicting. Please help me find Where am I wrong ?
Thank you
Let's consider a simple example. Let $G= \mathbb{Z}/3\mathbb{Z}=\{0,1,2\}$. Let $T=(1,2)$, which exchanges 1 with 2 and is an automorphism of $G$ of order 2. Take $x=1$, we have $T(x)=T ^{-1} (x)=2$, but $T(x ^{-1} )=T(2)=1$.