# Proving a subgroup of a Galois group is normal

"Let $$N/K$$ be a finite Galois extension with Galois group $$G = Gal(N/K)$$.

Let $$M$$ be an intermediate field of the extension, let $$E = Gal(N/M)$$, and let $$F = \cap_{\sigma \in G} σEσ^{-1}$$ .

Show that $$F$$ is a normal subgroup of $$G$$"

Where do I even begin here?

I know for all $$\sigma \in G$$, $$F \subset \sigma E \sigma ^{-1}$$, so $$\sigma ^{-1} F \sigma \subset E$$, so $$F$$ is a subgroup of $$E$$ under conjugation? Is this true?

Now how do I prove $$F$$ is normal?

• Have you shown that $F$ is normal in $G$? Nov 27 '18 at 22:08
• No, thats the bit I am struggling most with
– Dino
Nov 28 '18 at 5:10

This is just an exercise n group theory: take a group $$\;G\;$$ , a subgroup $$\;H\le G\;$$ and form what's called the core of $$\;H\;$$ , merely: the subgroup $$\;\bigcap_{x\in G}xHx^{-1}\;$$ . You can read about this in the web, but we can also do as follows:

Let $$\;X:=H\backslash G\;$$ be the set of left cosets of $$\;H\;$$ in $$\;G\;$$ , and define an action of $$\;G\;$$ on $$\;X\;$$ as follows: $$\;g\cdot xH:=(gx)H\;$$ . It's easy to check this is indeed an action. As usual, this action determines a homomorphism $$\;\phi:G\to Sym_X\cong S_n\;$$ , when $$\;n=[G:H]\;$$ , by the rule $$\;\phi g(xH):=\;(gx)H$$ (you only have to convince yourself that the above actually determines a permutation on $$\;Sym_X\;$$ for any $$\;g\in G$$ ).

Well, it is now a nice exercise to prove:

\begin{align*}(1)\;\;&\ker\phi=\bigcap_{x\in G}xHx^{-1}\\{}\\(2)\;\; &\ker\phi\;\;\text{is the maximal normal subgroup of \;G\; contained in H}\end{align*}

Give a try to the above two claims

• Let's assume we have no idea what the core of H is, so cannot use the other definitions of it. How would go about proving the question?
– Dino
Nov 28 '18 at 5:09
• @Dino Exactly as proposed above: show it is the kernel of a homomorphism and voila: you have it is normal! To show it is contained in $\;H\;$ is very easy (apply one element of the kernel to the left coset $\;H\;$ in $\;X\;$ ...) Nov 28 '18 at 8:21