TITLE: Which finite groups are generated by n involutions?
QUESTION [8 upvotes]: 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:

For each n, the smallest (abstract) n-polytope has an automorphism group that is isomorphic to the direct product of n copies of $C_2$, corresponding to the trivial Coxeter diagram on n nodes. So a finite group G cannot be the automorphism group of an abstract regular n-polytope for $n > \log_2(|G|)$. 
A (nontrivial) group generated by involutions has even order.
The abelianization of a group generated by involutions such that nonadjacent generators commute is a quotient of the group in (1), the direct product of n copies of $C_2$.

REPLY [14 votes]: Really quite a few finite simple groups are generated by three involutions, two of which commute (n=3).
For instance, this papers provides a (revised) proof that almost all sporadic groups have such a generating set:
Mazurov, V. D. "On the generation of sporadic simple groups by three involutions, two of which commute."
Sibirsk. Mat. Zh. 44 (2003), no. 1, 193–198; translation in Siberian Math. J. 44 (2003), no. 1, 160–164
MR1967616
DOI:10.1023/A:1022028807652
Its references provide a large list of other simple groups with this property:

almost all groups of lie type in char 2: MR1131150
almost all alternating groups: MR1172472
almost all groups of lie type in odd char: MR1454692 (low rank exceptions) and MR1601503 (all large rank)

Since one of the reviews was inaccurate, I quickly checked through the sources I had access to, and the following is probably quite close to accurate: Every finite simple group other than:

A6, A7, A8, S4(3)=U4(2), M11, M22, M23, McL
L3(q), U3(q) for all prime powers q
L4(q) for even prime powers q

has a generating set consisting of three involutions, two of which commute.  In particular the Monster group does have such a generating set (a short proof in the first paper, an earlier proof due to Simon Norton in a letter).
I (quickly) verified that A6, A7, A8, S4(3), M11, M22, M23 have no generating set of involutions where non-adjacent involutions commute (even for more than three generators).  You can bound n by the 2-rank of the group: having lots of commuting involutions means you have a large elementary abelian subgroup, and so n ≤ 5 for these groups.