1
$\begingroup$

For finite group $G$, Mobius function is defined on the lattice of subgroups of $G$ as: $\sum_{H\geq K} {\mu (H)}=\delta_{K,G}$. From this, $\mu (H)$ can be obtained by induction argument.

1. Can we define $\mu(H)$ explicitly (i.e. is there formula) for any $H \leq G$?

2. For cyclic groups of order $n$, does this Mobius function match with our number-theoretic Mobius function? How?

$\endgroup$