Take the 2-minute tour ×  One of the interesting problems in
abstract polytope theory is to determine, for a given finite
group, when that group is the automorphism group of a regular
abstract polytope. This is equivalent to the following question:
Given a finite group G, when is G generated by involutions
$\rho_0, \ldots, \rho_n$ such that $(\rho_i \rho_j)^2 = 1$ if
$|i - j| \geq 2$ and such that for all $I, J \subset \{0,
\ldots, n\}$ we have $\langle \rho_i \mid i \in I \rangle \cap
\langle \rho_i \mid i \in J \rangle = \langle \rho_i \mid i \in
I \cap J \rangle$?  The last property can be difficult to check,
so let's relax that requirement for now. If a finite group G is
generated by n involutions such that non-adjacent generators
commute, what can we say about the structure or size of G? Of
particular interest: what if G is simple?  Here are a few simple
observations:    1. For each n, the smallest (abstract)
n-polytope has an automorphism group that is isomorphic to the
direct product of n copies of $C_