# Subgroups of a Galois group $G$

I have the following exercise:

$L$ is a finite Galois extension of $K$ with Galois group $G$ (that is $G=Gal(L/K)$). Suppose $L_1$ and $L_2$ are subextensions and $G_1$ and $G_2$ are the respective subgroups of $G$. Show that $G$ is a direct product of $G_1$ and $G_2$ iff $L_1$ and $L_2$ are Galois extensions of $K$ s.t. $L_1 L_2 =L$ and $L_1 \cap L_2 =K$

What I am just wondering is the respective subgroups $G_1$ and $G_2$. If $G_1$ was Galois extension is $G_1=\operatorname{Gal}(L_1 /K)$ or is $G_1 =\operatorname{Gal}(L/L_1)$? I guessed it was the latter based on theorems for example the fundamental theorem of Galois extensions in the finite case but based on this. And is true that $G_1$ and $G_2$ are in general Galois extensions?

• The latter, yes. And $L/L_i$ is always a Galois extension. It's $L_i/K$ that may not be a Galois extension. – Dustan Levenstein May 30 '14 at 13:30
• $L_1$ is not Galois since its not normal Dustan. – Raxel May 30 '14 at 13:38
• @DustanLevenstein, what's the problem? In that case $\;L_1/\Bbb Q\;$ is not Galois... – DonAntonio May 30 '14 at 13:38
• Thanks, I missed that part of the assertion. – Dustan Levenstein May 30 '14 at 13:39
• Yes. There's no conflict with the result you linked to, because when $G$ is the direct product of $G_1$ and $G_2$, we have $G_1 \simeq G/G_2 = \operatorname{Gal}(L_2/K)$ and $G_2 \simeq G/G_1 = \operatorname{Gal}(L_1/K)$. – Dustan Levenstein May 30 '14 at 14:10

## 1 Answer

Hints (and you try to connect the dots): If $\;G\;$ is a group and $\;N_1\,,\,N_2\le G\;$ , then:

\begin{align*}\bullet&\;\;G=N_1\times N_2\iff\begin{cases}N_1,N_2\lhd G\\{}\\N_1N_2=G\\{}\\N_1\cap N_2=1\end{cases}\end{align*}

Now, using the Galois correspondence and theorems around this, and using your notation and your assumptions:

\begin{align*}\bullet&\;\;G=G_1G_2\iff L_1L_2=L\\{}\\\bullet&\;\;G_i\lhd G\iff L_i/K\;\;\text{is a normal extension}\iff L_i/K\;\;\text{is Galois}\\{}\\\bullet&\;\;G_1\cap G_2=1\iff L_1\cap L_2=K\;\text{(hint: what extension fits to the trivial sbgp.?)}\end{align*}

• I'm not familiar with that reversed play symbol. What does it mean in words? – Raxel May 30 '14 at 13:40
• @Raxel?? Do you mean $\;\lhd\;$ ? But that is the international symbol for normal subgroup: you must know basic group theory before engaging into Galois theory! – DonAntonio May 30 '14 at 13:41
• Well my lecture notes usually uses just words for it and Gallian does not use this symbol. – Raxel May 30 '14 at 13:42
• I don't know what Gallian book you're talking about, @Raxel, but if it is "Contemporary Abstract Algebra" he does use that symbol: 2nd edition, page 145, chapter 10. I really don't know any book in any language and after 1960 that doesn't use that symbol...not to mention that in a class at 2nd-3rd undergraduate level one usually knows already several books on some given subject. – DonAntonio May 30 '14 at 13:45
• Well, now you know one rather important symbol used, and it appears in each and every book in abstract algebra/ group theory as far as I recall, and it makes things way shorter and, imo, also simpler. Anyway, the answer's there and it doesn't change. – DonAntonio May 30 '14 at 14:07