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This must be known or easy for some of you, but here goes: Suppose $f_0,f_1:[n]\to [n]$ are invertible functions, where $[n]=\{0,\dots,n-1\}$ is a set of $n$ elements. For a word $w=w_1\dots w_m\in\{...

Let $G$ be a permutation group that acts on (say) $X=\{1,2,..,n\}$, and $H$ be a proper subgroup of $G$. Can one say anything precise about when the number of orbits of $H$ on $X$ will be equal to ...

Let $G$ be a permutation group on a set $X$ of points. How do we call a subset $S\subseteq X$ such that, for each permutation $\pi \in \mathrm{Sym}(S)$ there exists $g \in G$ acting like $\pi$ if ...

In this MO question it was mentioned that the following fact seems to be true: If $G$ is doubly transitive on $X$ and the one-point stabilizer $G_x$ has a non-trivial center, then $G$ is of ...