Permutation groups: is there a term for an arbitrarily-permuted subset of points? 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 restricted to $S$?
I would appreciate any comments, observations or notions related to such very special subsets of points.
My actual goal is searching for them algorithmically (say, for given generators). I work on a special kind of symmetries in graphs.
Thank you for any idea..
 A: Denote by $\newcommand{\eP}{\mathscr{P}}$ $\eP_G$ the  collection of subsets $S$ of $X$ with the property  you asked.  Observe that $\eP_G(X)$ is a simplicial collection of subses of $X$, i.e.,  if $S\in \eP_G(X)$ and $S'\subset S$, then $S'\in\eP_G(X)$.   It suffices to find the maximal elements of $\eP_G(X)$. 
Note that if $x_0\in X$, then $\{ x_0\}\in\eP_G(X)$. Moreover, if $x_0$ is is a fixed point of $G$, then $\{x_0\}$ is a maximal element of $\eP_G(X)$.  
Start with a set $S_1=\{x_1\}$. We will construct inductively  a maximal  subset of $\eP_G(X)$ containing $S_1$ as follows.
Suppose that you've  extended $S_1$ to  a subset $S_k=\{x_1,\dotsc,x_k\}\in\eP_G(X)$.  Then $S_k$ is not maximal if and only if there exists $x\in X\setminus S_k$ and $g\in G$ such that
$$ g(x_j)=x_j,\;\;\forall j=1,\dotsc,k-1, $$
$$ g(x_k)=x,\;\;g(x)=x_k. $$
Indeed, if such a pair $(x,g)$ existed, then the collection $\DeclareMathOperator{\Sym}{Sym}$ $\Sym(S_k)\cup\{g\}$ would generate $\Sym(S_k\cup\{x\})$.
It suffices to look for such a $g$  only by searching in the subgroup  $\DeclareMathOperator{\Fix}{Fix}$
$$ \Fix_{x_1,\dotsc, x_{k-1}}:=\Fix_{x_1}\cap\cdots \cap \Fix_{x_{k-1}}, $$
where 
$$\Fix_x:=\big\{\; g\in G;\;\;gx=x\;\big\}. $$
You need to find  $g\in \Fix_{x_1,x_2,\dotsc, x_{k-1}}$  such that $gx_k\neq x_k$ and $g^2x_k=x_k$. If no such $g$ exists, then $S_k$ is maximal.
Remark. The collection $\eP_G(X)$ produces an invariant  of the group $G$. Denote by $\Delta(G, X)$ the simplicial complex associated to $\eP_G(X)$. More precisely, to any subset $S\in\eP_G(X)$ there is an assiciated $|S|-1$-dimennsional simplex of $\Delta(G,X)$. 
The homology of this simplicial complex is an invariant of the action of $G$ on $X$. In particular, if $X=G$ and $G$ acts on itself by left translations, then we obtain a simplicial complex $\Delta(G)=\Delta(G,G)$. It is an invariant of the group and so is its homology $H_\bullet\big(\Delta(G)\big)$. 
Note that if $G$ Abelian, then $\dim \Delta(G)\leq 1$ so that $\Delta(G)$ is a graph. There are many interesting questions that one can ask.  For example.  


*

*Are there any non-Abelian groups $G$ such that $\dim\Delta(G)\leq 1$? 

*What kind of graphs have the form $\Delta(G)$ for some group $G$?


Note that when $G$  is Abelian, the graph $\Delta(G)$ has  simple description.


*

*It has one vertex  for every $g\in G$.

*Two distinct vertices $g_1,g_2$ are connected by an edge if and only if $g_2g_1^{-1}$ is an involution. 


Thus the graph $\Delta(G)$ is connected if and only if every element of $G$ is a product of involutions. Moreover, if $i(G)$ is the number of   (nontrivial) involutions of $G$.  The Euler  characteristic of $\Delta(G)$ is
$$ \chi\big(\Delta(G)\big)=|G|\left( 1-\frac{i(G)}{2}\right).  $$
