The cohomology of the Mathieu group M23

and, as a result of our calculation Hi(M23;Z) = 0 for i < 5. In particular M23 is the first known counterexample to the conjecture that if G is a finite group with Hi(G;Z) = 0, i = 1, 2, 3, then G = {1}‡. (M11, the first Mathieu group and J1, the first Janko group also satisfy Out(G) = Mult(G) = 1, but for both of these groups H3(G;Z) = 0.) It would be tempting to amend the conjecture. It is very likely that it only fails for a very small number of the sporadics among the simple groups. So one might well suspect that there is a (small) finite number n so that H(G;Z) = 0 for 0 < i ≤ n implies that G = {1} if G is finite. But I have no idea as to a suitable candidate for n.