Elementary Proof of a Theorem of Hawkes, Isaacs and \"Ozaydin

We present an elementary proof of the theorem of Hawkes, Isaacs and \"Ozaydin, which states that $\Sigma\,\mu_{G}(H,K)\equiv 0$ mod $d$, where $\mu_{G}$ denotes the M\"obius function for the subgroup lattice of a finite group $G$, $H$ ranges over the conjugates of a given subgroup $F$ of $G$ with $[G:F]$ divisible by $d$, and $K$ over the supergroups of the $H$ for which $[K:H]$ divides $d$. We apply the theorem to obtain a result on the number of solutions of $|\langle H,g\rangle|\mid n$, for said $H$ and a natural number $n$. The present version of the article includes an additional result on a quantity studied by K.S. Brown.