Hypotheses:
- $G$ is a finite group
- $A$ and $B$ are subgroups if $G$.
Definitions:
- $AB = \{ ab \mathbin{|} a \in A, b \in B \}$ is the product of $A$ and $B$
- $A \vee B$ is the join of (subgroup of $G$ generated by) $A$ and $B$
- $A \wedge B$ is the intersection $A$ and $B$ (a subgroup of $G$)
It is known that $|AB|$ is equal to $|A||B| \mathbin{/} |A \wedge B|$, and that $|AB|$ is less than or equal to $|A \vee B|$.
Question: Is $|AB|$ a divisor of $|A \vee B|$ ? If so, I would like to have a proof. If not, I would like to have a concrete counter-example.
UPDATE (Galois theoretic motivation)
Let $L$ be a normal extension of the rationals, and let $G$ be its
Galois group. Let $\alpha$ be some element of $L$, generating a
subfield fixed by the subgroup $A$ of $G$. The degree of the minimal
polynomial $f(x)$ of $\alpha$ over $Q$ is then $[G : A]$. Consider now
the subfield $K$ of $L$ fixed by some subgroup $B$ of $G$. The minimal
polynomial $g(x)$ of $\alpha$ over $K$ is a divisor of $f(x)$ and has
degree $[B : A \wedge B]$. In particular, the degree of $g(x)$ is a
divisor of the degree of $f(x)$, that is, $[B : A \wedge B]$ is a
divisor of $[G : A]$.
Now, for any finite group $G$ and subgroups $A$ and $B$ holds: $$ 1 = \frac{|AB||A \wedge B|}{(|A||B|)} \leq \frac{|A \vee B||A \wedge B|}{(|A||B|)} \leq \frac{|G||A \wedge B|}{(|A||B|)} = \frac{[G : A]}{[B : A \wedge B]} $$
The full question would thus be whether all these values are integers or not, considering the case that $G$ is the Galois group of some normal extension of the rationals as well as the case that $G$ is an arbitrary finite group.
What is known:
If $G$ is the Galois group of a normal extension of the rationals, then $[G : A] \mathbin{/} [B : A \wedge B]$ is an integer.- $|AB|$ is not necessarily a divisor of $|A \vee B|$ (answer to this question)
UPDATE (wrong claim)
I did a mistake: $g(x)$ is a divisor of $f(x)$, but the degree of $g(x)$ is not necessarily a divisor of the degree of $f(x)$. This invalidates the question, I was on a wrong path.
