Is each element of a permutation group acts like a ring homomorphism? See the "conjecture" The following is an exercise from A First Course in Abstract Algebra - JB Fraleigh :

One way I guess to solve the above exercise may be this approach that if $\tau \in G(K/F)$ be arbitrary and if $\tau (N) = N$ and $\tau (T) = T$ then $N,T \in F$. We use here the fact that $G(K/F)$ is a group so any $\tau$ on all of its element just re-orders the Galois group. But this requires the following :
Conjecture : $\tau (\sigma_1(\alpha) \sigma_2(\alpha) \dots \sigma_n(\alpha))=\tau (\sigma_1(\alpha)) \tau(\sigma_2(\alpha)) \dots \tau(\sigma_n(\alpha))$ and $\tau (\sigma_1(\alpha) + \sigma_2(\alpha) + \dots + \sigma_n(\alpha))=\tau (\sigma_1(\alpha)) + \tau(\sigma_2(\alpha)) + \dots + \tau(\sigma_n(\alpha))$.
If the conjecture above is true how to prove it? If not what is the correct approach to solve the main exercise?
 A: As far as I can see your conjecture is correct but the title is misleading: the element $\tau$ acts as a ring homomorphism because it lies in the Galois group (which consists of ring homomorphisms by definition). If $\tau$ would be an element of a random permutation group (as the title suggest) we could not assume that it acts as  a ring homomorphism - in fact we would have no idea what it does with elements of $K$ at all.
So your conjecture is correct, but as Arturo hints at in the comments, that in itself does not yet constitute a complete proof. You also need a separate argument that $\tau (\sigma_1(\alpha)) \tau(\sigma_2(\alpha)) \dots \tau(\sigma_n(\alpha))$, which appears in the right hand side of your conjecture, equals the original element $\sigma_1(\alpha)\sigma_2(\alpha) \dots \sigma_n(\alpha)$. (And similar for the version with + signs.)
But perhaps you already had this part and just did not type it out.
UPDATE: in rereading I see you did type it out, but earlier. It is where you say 'just re-orders the Galois group'. So yeah, on second look I'd say your proof is correct AND complete, maybe just add something like 'since $\tau \in G(K/F)$ it acts by ring homomorphisms' just to remind your future self.
