Given a finite simple group $G$, we can consider the quasisimple extensions $\tilde G$ of $G$, that is to say central extensions which remain perfect. Some basic group cohomology (based on the standard trick of averaging a cocycle to try to make it into a coboundary) shows that up to isomorphism, there are only finitely many such quasisimple extensions, and they are all quotients of a maximal quasisimple extension, which is known as the universal cover of $G$, and is an extension of $G$ by a finite abelian group known as the Schur multiplier $H^2(G,{\bf C}^\times)$ of $G$ (or maybe it would be slightly more accurate to say that it is the Pontryagian dual of the Schur multiplier, although up to isomorphism the two groups coincide).

On going through the list of finite simple groups it is striking to me how small the Schur multipliers are for all of them; with the exception of the projective special linear groups $A_{n-1}(q)=PSL_n({\bf F}_q)$ and the projective special unitary groups ${}^2 A_{n-1}(q^2) = PSU_n({\bf F}_q)$, all other finite simple groups have Schur multiplier of order no larger than 12, and even the projective special linear and special unitary groups of rank $n-1$ do not have Schur multiplier of size larger than $n$ (other than a finite number of small exceptional cases, but even there the largest Schur multiplier size is 48). In particular, in all cases the Schur multiplier is much smaller than the order of the group itself (indeed it is always of order $O(\sqrt{\frac{\log|G|}{\log\log|G|}})$). For comparison, the standard proof of the finiteness of the Schur multiplier (based on showing that every $C^\times$-valued cocycle on $G$ is cohomologous to $|G|^{th}$ roots of unity) only gives the terrible upper bound of $|G|^{|G|}$ for the order of the multiplier.

In the case of finite simple groups of Lie type, one can think of the Schur multiplier as analogous to the notion of a fundamental group of a simple Lie group, which is similarly small (being the quotient of the weight lattice by the root lattice, it is no larger than $4$ in all cases except for the projective special linear group $PSL_n$, where it is of order $n$ at most). But this doesn't explain why the Schur multipliers for the alternating and sporadic groups are also so small. Intuitively, this is asserting that it is very difficult to make a non-trivial central extension of a finite simple group. Is there any known explanation (either heuristic, rigorous, or semi-rigorous) that helps explain why Schur multipliers of finite simple groups are small? For instance, are there results limiting the size of various group cohomology objects that would support (or at least be very consistent with) the smallness of Schur multipliers?

Ideally I would like an explanation that does not presuppose the classification of finite simple groups.

nonabelian tensor product of groups, see the bibliography at pages.bangor.ac.uk/~masoao/nonabtens.html – Ronnie Brown May 17 '13 at 20:33