Which finite groups are generated by n involutions? - MathOverflow

Which finite groups are generated by n involutions?

2010-05-28T18:35:14Z

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 $C_2$.

Answer by Bruce Westbury for Which finite groups are generated by n involutions?

2010-05-28T18:51:59Z

Have you read "Regular Polytopes" by H.S.M. Coxeter? or any other text book on reflection groups or Coxeter groups?

In a nutshell, the strategy is to write down the Gram matrix of an inner product that is preserved by the reflections and so by the group generated by the reflections. Then the group is finite if and only if this inner product is positive definite.

Answer by Jack Schmidt for Which finite groups are generated by n involutions?

2010-05-28T19:17:31Z

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.

Answer by Michael Hartley for Which finite groups are generated by n involutions?

2011-04-05T08:19:39Z

Gabe, Dimitri Leemans does a lot of work on this kind of question - have you got all his stuff?