3 problems

Let $H$ be a finite group, and let $h_{\rm OD}(H)$ denote the number of isomorphism classes of groups $G$ such that $|G|=|H|$ and ${\rm D}(G)={\rm D}(H)$. A group is OD-characteriz…

Let $G$ be a finite group, and let $S$ be a non-abelian simple group. Write $\mu(G)$ for the set of proportions of $r$-singular elements of $G$ as $r$ ranges over the primes dividi…

Same-order type characterization conjecture. Then $G\cong S$.