I have been studying Derek Holt's Handbook of Computational Group Theory. I am looking at Proposition 4.7, and I can't figure out what I'm doing wrong.
Proposition 4.7(i) states that, given $K$, a subgroup of a permutation group $G$, and a base $B$ for $G$, an element $g \in G$ is the $\prec$-least element of its coset $gK$ if and only if $B_j^g$ is the $\prec$-least element of its orbit under $K_{B_1^g, \dots, B_{j-1}^g}$, where $B_i$ is the $i$th element of $B$, and $K_{B_1^g, \dots, B_{j-1}^g}$ is the stabilizer of $B_1^g, \dots, B_j^g$ in $K$. I did an example myself where $G = S_4$, $K = S_3$, and $B = [1, 2, 3]$.
Looking at the coset:
$(1\ 4) - [4,2,3]\\ (1\ 2\ 4) - [2,4,3]\\ (1\ 3\ 4) - [3,2,4]\\ (2\ 3)(1\ 4) - [4,3,2]\\ (1\ 2\ 3\ 4) - [2,3,4]\\ (1\ 3\ 2\ 4) - [3,4,2]$
It is clear the element $g = (1\ 2\ 3\ 4) - [2,3,4]$ is the least element under $\prec$. However, $B_1^g = 2$, and $2$ is not the least element of the orbit of $2$ in $K$ (since $j=1$ we don't stabilize anything). I feel like the proposition is stating something much more obvious than I'm making it. Perhaps the orbit is restricted to the coset?