Follow up of my previous doubt (Finite groups as subgroups of dihedral groups) , can anyone tell me is that fact correct or not, that any finite group can be realized as a subgroup of the group of symmetries of a regular polytope?

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    $\begingroup$ A more interesting and harder question is whether any finite group can be realized as exactly the group of symmetries of a regular polytope. $\endgroup$ – Qiaochu Yuan Sep 9 '16 at 19:38
  • $\begingroup$ @QiaochuYuan I don't consider myself an exert in group theory. So it will be very helpful if you give me some reference. Actually this problem came to my mind while I was trying to solve a problem in mapping class group. $\endgroup$ – Anubhav Mukherjee Sep 9 '16 at 20:09
  • $\begingroup$ @QiaochuYuan you probably didn't mean to include the word "regular", as that's not a very interesting question as stated... $\endgroup$ – Nate Sep 10 '16 at 14:38
  • $\begingroup$ @Nate: why not? $\endgroup$ – Qiaochu Yuan Sep 10 '16 at 16:33
  • $\begingroup$ There aren't that many regular polytopes, and their symmetry groups are known. Above dimension $4$ there are only $3$ regular polytopes of each dimension, and their symmetry groups are symmetric and hyperoctahedral groups. $\endgroup$ – Nate Sep 10 '16 at 17:49

A simplex has a symmetric group as its symmetry group, and every finite group can be embedded in a symmetric group.

  • $\begingroup$ can you give me some reference where I can read this proof? $\endgroup$ – Anubhav Mukherjee Sep 9 '16 at 19:13
  • $\begingroup$ A simplex is an object in the family: point, line segment, triangle, tetrahedron, pentachoron... The usual definition of a simplex is something like "the convex hull of an affinely independent collection of points" (the points are called the vertices of the simplex). This definition is "symmetric" with respect to the vertices: it's obvious that every affine symmetry of a simplex permutes the vertices, and not hard to see that every permutation of the vertices gives a unique affine symmetry. So the group of (affine) symmetries is just the group of permutations of the vertices. $\endgroup$ – Robin Saunders Sep 9 '16 at 23:21
  • $\begingroup$ You can define a simplex combinatorially instead: its vertices, or 0-cells (since they are 0-dimensional), are an arbitrary (finite) collection of points, and the k-cells correspond to the subcollections of k+1 vertices. Again, this definition is "symmetric" in the vertices and so there must be at least one symmetry for each permutation of the vertices; but the images of the rest of the cells are fixed by the images of the vertices. $\endgroup$ – Robin Saunders Sep 9 '16 at 23:24
  • $\begingroup$ @Anubhav One can realize a simplex as the basis coordinate vectors in $\mathbb{R}^n$. One uses them to create a polytope with $n$ vertices. Any symmetry is determined by the permutation it induces on the set of vertices, and every permutation is realizable as a symmetry (just use the associated permutation matrix as a transformation of space). $\endgroup$ – arctic tern Sep 10 '16 at 2:13

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