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Let $M$ by an $m$-dimensional symmetric space (or a general Riemannian manifold). The finite distinct points $p_1,p_2,\cdots,p_n\in M$ are said symmetric, if for any permutation $\sigma$ on $1,2,\cdots,n$, there exists an isometry $f$ of $M$ such that $f(p_i)=p_{\sigma(i)}$ for all $i=1,2,\cdots,n$.

Question 1: Given $M$, for example, $M$ is the $m$-sphere, what is the largest integer $n$ such that there exists symmetric points $p_1,\cdots, p_n\in M$?

Question 2: Given $n$, how to determine all possible positions of symmetric $n$-points on $M$?

Question 3: Are there any backgrounds, motivations & related references for symmetric points on symmetric spaces?

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    $\begingroup$ You are basically asking whether there is a copy of the symmetric group $S_n$ on $n$ elements in Isom(M). Observe that your condition implies that for any subset of $n-2$ points there exists some totally geodesic submanifold $N$ that contains all of them. On the $m$-sphere I think this may be enough to say that the max is $n = m+2$ arranged in the shape of the regular $n$-simplex. $\endgroup$ Sep 18, 2015 at 4:48

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