Let $L$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$. The Möbius function $\mu$ for $L$ is defined recursively by: for $\forall a,b \in L$ with $a<b$, $\mu(b,b) = 1$ and $\mu(a,b) = -\sum_{a<c\le b}\mu(c,b)$.

The Möbius number of $L$ is defined by $\mu(\hat{0},\hat{1})$.

Define the Möbius number $\mu(G)$ of a finite group $G$ to be the Möbius number of its subgroup lattice. For the nonabelian finite simple groups of small order, we observe that $$\mu(G) \in |G| \mathbb{Z}$$

**Question**: Is it true for any nonabelian finite simple group?

*Remark:* It is true if $|G| < 100000$ (see the table below).

It's false for nonsimple groups because for $p$ (odd) prime, $\mu(S_p) = p!/2$ ([S]) and $\mu(C^2_p) = p$.

Table for the nonabelian finite simple groups of order $< 100000$: $$ \begin{array}{c|c|c|c|c|c} G & |G| & \mu(G) & \mu(G)/|G| & |Out(G)| & \newline \hline A_5 & 60 & -60 & -1 & 2 & \newline \hline PSL(2,7) & 168 & 0 & 0 & 2& \newline \hline A_6 & 360 & 720 & 2 & 4 & \newline \hline PSL(2,8) & 504 & -504 & -1 & 3 \newline \hline PSL(2,11) & 660 & 660 & 1 & 2 \newline \hline PSL(2,13) & 1092 & -1092 & -1 & 2 \newline \hline PSL(2,17) & 2448 & 0 & 0 & 2 \newline \hline A_7 & 2520 & 2520 & 1 & 2 \newline \hline PSL(2,19)& 3420 & 3420 & 1 & 2 \newline \hline PSL(2,16)& 4080 & 0 & 0 & 4 \newline \hline PSL(3,3)& 5616 & 0 & 0 & 2 \newline \hline PSU(3,3)& 6048 & 0 & 0 & 2 \newline \hline PSL(2,23)& 6072 & 0 & 0 & 2 \newline \hline PSL(2,25)& 7800 & 0 & 0 & 4 \newline \hline M_{11} & 7920 & -7920 & -1 & 1 \newline \hline PSL(2,27)& 9828 & 9828 & 1 & 6 \newline \hline PSL(2,29) & 12180 & 12180 & 1 & 2 \newline \hline PSL(2,27)& 14880 & 29760 & 2 & 2 \newline \hline A_8& 20160 & 20160 & 1 & 2 \newline \hline PSL(3,4)& 20160 & -120960 & -6 & 12 \newline \hline PSL(2,37)& 25308& -25308& -1 & 2 \newline \hline PSp(4,3)& 25920& -25920& -1 & 2 \newline \hline Sz(8)& 29120& -29120& -1 & 3 \newline \hline PSL(2,32)& 32736& -32736& -1 & 5 \newline \hline PSL(2,41)& 34440& 68880& 2 & 2 \newline \hline PSL(2,43)& 39732& -39732& -1 & 2 \newline \hline PSL(2,47)& 51888& 0& 0 & 2 \newline \hline PSL(2,49)& 58800& 117600& 2 & 4 \newline \hline PSU(3,4)& 62400& 0& 0 & 4 \newline \hline PSL(2,53)& 74412& -74412& -1 & 2 \newline \hline M_{12} & 95040 & 95040 & 1 & 2 \newline \end{array}$$