I am reading "Algebraic Graph Theory" by Norman Biggs (1974), and I have problem understanding the following definition and proposition.
We define a stabilizer sequence as follows:
A stabilizer sequence of the $t$-arc $(a_0,a_1,\ldots,a_t)$ in a graph $X$ is the sequence $Aut, F_t, F_{t-1},\ldots,F_1,F_0$ of subgroups of $Aut(X)$, where $F_i \; (0 \leq i \leq t)$ is defined to be the pointwise stabilizer of the set $\{a_0,a_1,\ldots,a_{t-i}\}$.
I understand that a stabilizer sequence is a sequence of subgroups, where each element in the sequence fixes some set of vertices in the graph $X$, and each element in the sequence fixes more than the previous.
Now the following is stated, without a proof:
Since $Aut(X)$ is transitive of $s$-arcs $(1 \leq s \leq t)$ it follows that all stabilizer sequences of $t$-arcs are conjugate in $Aut(X)$.
I also know the definition and (and at least some of) the intuition behind conjugacy, but I do not see why these sequences of stabilizers, are conjugate.
If some definition is missing, or more information about the general setting is needed, there might be information in another question I have asked earlier: Proof about cubic $t$-transitive graphs.