Let $X$, $Y$ be smooth, connected, compact manifolds (for instance, projective varieties) and $f \colon X \longrightarrow Y$ be a finite, branched cover of degree $n$, with branch locus $B \subset Y$. We can then associate to $f$ its monodromy representation $$\theta\_f \, \colon \pi\_1(Y-B) \longrightarrow S\_n,$$ so that $\mathrm{im} \, \theta\_f$ is a transitive subgroup of $S\_n$. Conversely, isomorphism classes of connected covers of $Y$, branched over $B$, are in bijection to monodromy representations of the type above, up to conjugacy in $S\_n$.


We now assume that $f$ is *simple*, that is that for all $b \in B$ the fibre $f^{-1}(b)$ consists of exactly $n-1$ points, and we call $f$ an *elementary* cover if it is not possible to factor it as $$X \stackrel{g}{\longrightarrow} Z \stackrel{h}{\longrightarrow} Y,$$
where $g\colon X \longrightarrow Z$ is a branched cover and $h \colon Z \longrightarrow Y$ is an ordinary (i.e, unramified) cover.



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> **Q.** What is the characterization of simple, elementary covers $f \colon X \longrightarrow Y$, branched over $B$, in terms of their monodromy representation $\theta\_f$?
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Monodromy representation of elementary simple covers
I am just writing my comments as an answer. For every subgroup $H$ of the symmetric group $\mathfrak{S}\_n$, define a relation on $\{1,\dots,n\}$ by $a\sim b$ if either $a$ equals $b$ or if the transposition $(a,b)$ is contained in $H$. By definition, this is symmetric and reflexive. Let $a,b,c$ be pairwise distinct elements of $\{1,\dots,n\}$ such that $a\sim b$ and $b\sim c$. Since $H$ contains $(a,b)$ and $(b,c)$, and since $H$ is a subgroup, also $H$ contains $$(a,c) = (a,b)\circ (b,c) \circ (a,b).$$ Thus, also $a\sim c$. So this is an equivalence relation. The associated partition of $\{1,\dots,n\}$ is preserved by $H$. 


Therefore, every subgroup of $H$ that contains at least one transposition and that preserves no nontrivial partition must equal all of $\mathfrak{S}\_n$. Geometrically, this says that a simple, elementary cover with nonempty branch locus has monodromy group equal to the full symmetric group. If the branch locus is empty, this can fail, e.g., it fails for unbranched cyclic covers.


