I am following the textbook Representation Theory of the Symmetric Groups, by Tullio C.-S., Fabio S., and Filippo T., and am confused at the definition of a Gelfand pair.
The definition is:
Let $G$ act transitively on a set $X$. Fix $x_0 \in X$ and denote its stabilizer by $K=\{g \in G : g x_0 = x_0\}$. We can identify $X$ with the space of right cosets of $K$ in $G$ $(X = G/K)$. If the permutation representation $\lambda$ of $G$ has a decomposition that is multiplicity-free, then $(G,K)$ is a Gelfand pair.
Here the permutation representation is the representation of $G$ on $L(X)$, the space of complex functions on $X$, where:
$$[\lambda (g)f](x) = f(g^{-1}x)$$
I don't understand why $K$ is necessary in the definition. We can identify $X$ with $G/K$, but the action works on $X$ just the same. Doesn't this mean that all the stabilizers on $X$ form a Gelfand pair with $G$? What's the point of mentioning $K$ at all?