Let $L/K$ be an extension of number fields, and $M/K$ the Galois closure of $L/K$ (everything happens inside a suitably large characteristic zero field $\Omega$). Let $p$ be a discrete prime of $K$.


Given the group $G={\rm Gal}(M/K)$, its subgroup $H={\rm Gal}(M/L)$, and the splitting of $p$ in $L/K$ can one find the splitting of $p$ in $M/K$? That is, can one find the ramification index $e$ and inertial degree $f$ of a prime $P$ of $M$ lying above $p$?




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Turning the above question into a group-theoretic one, I got the following: let $D$ be a finite group, and $X$ a finite set on which $D$ acts faithfully on the right. Can we obtain the order $d$ of $D$ knowing the sizes of all the $D$-orbits in $X$?


(To switch to this question from the original one, look at the natural right action of a decomposition group (resp. an inertia group) $D$ at $P$ on the coset space $X=H\backslash G$. The fact that $M$ is the Galois closure of $L$ ensures that $D$ acts on $X$ faithfully.)



Decomposition of primes in Galois closures of number fields
**Changed answer:** The answer is *no*: Let $L$ be a root field of $X^3-2$ over $\mathbb Q$, and $M$ the Galois closure of $L/\mathbb Q$. Then the primes $2$ and $3$ are both totally ramified in $L$, yet in both cases there is only one prime $P$ above $p=2$ or $p=3$, with $e(P)=3$ if $p=2$, and $e(P)=6$ if $p=3$.


