Möbius function of subgroup lattice (reviewed)

Given a finite group $G$, we define the möbius subgroup function $\mu_G(H)$ on the lattice of subgroups of $G$ by setting $\mu_G(G) = 1$ and then inductively using the relation $$\mu_G(H) = -\sum_{H' > H} \mu_G(H').$$

Similarly, we define the möbius quotient function $\nu_G(N)$ on the lattice of normal subgroups of $G$ by setting $\nu_G(1) = 1$ and inductively using the relation $$\nu_G(N) = -\sum_{N' < N} \nu_G(N').$$

The möbius subgroup function is used in computing the rank of $G$, while the möbius quotient function is used in computing the number of faithful representations of $G$.