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]](http://www.sciencedirect.com/science/article/pii/S009731659792762X)) and $\mu(C^2\_p) = p$.




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Table for the nonabelian finite simple groups of order $< 100000$:
$$ \begin{array}{c|c|c|c|c|c}
G & |G| & \mu(G) & \mu(G)/|G| & |Out(G)| & \newline 
 \hline 
 A\_5 & 60 & -60 & -1 & 2 & \newline 
 \hline 
 PSL(2,7) & 168 & 0 & 0 & 2& \newline 
 \hline
 A\_6 & 360 & 720 & 2 & 4 & \newline 
 \hline
 PSL(2,8) & 504 & -504 & -1 & 3 \newline 
 \hline
PSL(2,11) & 660 & 660 & 1 & 2 \newline 
 \hline
PSL(2,13) & 1092 & -1092 & -1 & 2 \newline 
 \hline
 PSL(2,17) & 2448 & 0 & 0 & 2 \newline 
 \hline
 A\_7 & 2520 & 2520 & 1 & 2 \newline 
 \hline
 PSL(2,19)& 3420 & 3420 & 1 & 2 \newline 
 \hline
 PSL(2,16)& 4080 & 0 & 0 & 4 \newline 
 \hline
 PSL(3,3)& 5616 & 0 & 0 & 2 \newline
 \hline
 PSU(3,3)& 6048 & 0 & 0 & 2 \newline
 \hline
 PSL(2,23)& 6072 & 0 & 0 & 2 \newline
 \hline
 PSL(2,25)& 7800 & 0 & 0 & 4 \newline
 \hline
 M\_{11} & 7920 & -7920 & -1 & 1 \newline
 \hline
 PSL(2,27)& 9828 & 9828 & 1 & 6 \newline
 \hline
 PSL(2,29) & 12180 & 12180 & 1 & 2 \newline
 \hline
 PSL(2,27)& 14880 & 29760 & 2 & 2 \newline
 \hline
 A\_8& 20160 & 20160 & 1 & 2 \newline
 \hline
 PSL(3,4)& 20160 & -120960 & -6 & 12 \newline
 \hline
 PSL(2,37)& 25308& -25308& -1 & 2 \newline
 \hline
PSp(4,3)& 25920& -25920& -1 & 2 \newline
 \hline
Sz(8)& 29120& -29120& -1 & 3 \newline
 \hline
PSL(2,32)& 32736& -32736& -1 & 5 \newline
 \hline
PSL(2,41)& 34440& 68880& 2 & 2 \newline
 \hline
PSL(2,43)& 39732& -39732& -1 & 2 \newline
 \hline
PSL(2,47)& 51888& 0& 0 & 2 \newline
 \hline
PSL(2,49)& 58800& 117600& 2 & 4 \newline
 \hline
PSU(3,4)& 62400& 0& 0 & 4 \newline
 \hline
 PSL(2,53)& 74412& -74412& -1 & 2 \newline
 \hline
M\_{12} & 95040 & 95040 & 1 & 2 \newline
\end{array}$$



The Möbius number of the nonabelian finite simple groups
The answer to your question is ``yes". Something more general was proved by Hawkes, Isaacs and \"Ozaydin in a 1989 paper in the Rocky Mountain Journal of Mathematics.


CORRECTION: As Sebastien Palcoux notes below, this result is due to Kratzer and Th\'evenaz


More precise results for the groups $PSL\_2(p)$ were known to Philip Hall, and $PSL\_2(q)$ was handled by Martin Downs, as were the Suzuki groups.


Some results on symmetric groups appear in a paper of mine, and other classes of ``small" almost simple groups are addressed in my thesis,


<http://www.math.wustl.edu/~shareshi/thmsri.ps>


(Sorry, it is a postscript file and some non-mathematical pages are repeated.) 


