# Elements of $Gal(K(\alpha_1, \cdots, \alpha_n): K)$ which send a specific subset of $\alpha_1, \cdots, \alpha_n$ to another specific subset

Suppose $$L:K$$ is a splitting field extension for a monic, separable polynomial $$f \in K[t]$$ which is irreducible over $$K$$. Let $$\alpha_1, \cdots, \alpha_n$$ be the distinct roots of $$f$$ in $$L$$. Let $$k < n/2$$ and choose a subset $$\beta_1, \cdots, \beta_k, \delta_1, \cdots, \delta_k \subset \alpha_1, \cdots, \alpha_n$$ of the roots (all the elements are distinct.)

I know that the maps in $$Gal(L:K)$$ permute the roots $$\alpha_1, \cdots, \alpha_n$$. Can I always find an element $$\tau \in Gal(L:K)$$ such that $$\tau(\beta_1) = \delta_1, \cdots, \tau(\beta_k) = \delta_k$$?

If not, how do I know which subsets of $$\alpha_1, \cdots, \alpha_n$$ can be mapped to which other subsets? (Is there a way of knowing this, in general?)

The short answer is no. Consider the cylotomic field extension $$\mathbb{Q}(\zeta_p)$$ over $$\mathbb{Q}$$ with Galois group $$\mathbb{Z}_p^{\times}$$, and basis $$\zeta_p, ..., \zeta_p^{p-1}$$. Then there is no automorphism that will send $$\zeta_p$$ to $$\zeta_p^2$$ and $$\zeta_p^3$$ to $$\zeta_p^4$$.
In general this depends on the structure of the Galois group. A permutation group on $$n$$ elements is called $$k$$-transitive if any ordered $$k$$-tuple can be mapped to any other ordered $$k$$-tuple (which is exactly what you have asked). Multiply transitive groups (specifically doubly transitive groups) are rare - all doubly transitive groups are known. There is a list of classifications here http://mathworld.wolfram.com/TransitiveGroup.html
There are extensions for which this is possible. For instance, Hilbert showed that $$S_n$$ and $$A_n$$ can represented as Galois groups of Galois extensions of $$\mathbb{Q}$$. (It is a nice exercise to check that $$S_n$$ is $$n$$-transitive and $$A_n$$ is $$n-2$$-transitive).