Definition of Galois Group I'm revising for a module on Galois Theory and have trouble understanding the definition for a Galois group of a field extension $K:F$.
Define the Galois group of $K:F$ as $\Gamma(K:F)=\{\sigma \in \operatorname{Aut}(K)|\forall a\in F, \sigma(a)=a, F\leq \operatorname{Fix}(\sigma)\}$
I can't quite understand the point of that last condition, $F\leq \operatorname{Fix}(\sigma)$.
Also, the next statement after the definition is: Clearly, $F \leq \operatorname{Fix}(\Gamma(K:F))$. 
Why is this so?
 A: The fundamental tool in Galois theory is the Galois connection induced by the binary relation 'fixing' between automorphisms and elements of a given field.
In other words, write e.g. $\newcommand\f{\,\mathtt f\,} \alpha \f x$ for $\alpha(x)=x$, if $\alpha:K\to K$ automorphism, and $x\in K$.
Now the given notations, by definition.
$$Fix(\sigma)=\{a\in K\,\mid\, \sigma\f a\} $$
So, $F\le Fix(\sigma)$ means exactly that all elements of $F$ are fixed by $\sigma$. Suppose that $K$ is fixed.
$$\Gamma(K:F)=\{\sigma\,\mid\,F\le Fix(\sigma)\}\,.$$
Now the assignments $\Phi:S\mapsto \{\sigma\in Aut(K)\,\mid\, \forall s\in S: \sigma\f s\}$ and $\Psi:H\mapsto \{a\in K\,\mid\,\forall\sigma\in H: \sigma\f a\}$ map any subsets $S\subseteq K$ and $H\subseteq Aut(K)$ to a subgroup and a subfield, respectively.
Now we have $\Gamma(K:F)=\Phi(F)$ and $Fix(H)=\Psi(H)$, and we want to prove $F\subseteq \Psi(\Phi(F))$, but this is immediate from the definitions, as for an $s\in F$, we have $\sigma\f s$ for all $\sigma\in\Phi(F)$.
In general, try to prove (for arbitrary binary relation $\f$ between aribtrary two sets) that both $\Phi$ and $\Psi$ reserves the order given by inclusion, hence both compositions $\Phi\circ\Psi$ and $\Psi\circ\Phi$ are monotonic,
 and that already $\Phi\circ\Psi\circ\Phi=\Phi$ and $\Psi\circ\Phi\circ\Psi=\Psi$. 
