10 problems

Shi's conjecture. Each finite simple group $G$ is determined by its order $|G|$ and its spectrum $(G)$.

The h-value conjecture.

Let $G$ be a group and $S$ be a finite simple group. Let $pe(H)$ denote the invariant used in the source, namely the combined data of the relevant element-order counts. **NPE char…

Let $S$ be a nonabelian simple group, let $G$ be a group, and let $p$ be an odd prime divisor of $|S|$. For a group $H$, let $H(2)$ and $H(p)$ denote the sets of elements of orders…

Let $G$ and $S$ be finite nonabelian simple groups, and let $G(2)$ and $S(2)$ denote their sets of elements of order $2$. Herzog's conjecture. If … then … The conjecture is false i…

Let $S$ be a finite simple group, and let $p$ be the largest prime divisor of $|S|$. For a group $H$, let $H(p)$ denote the set of elements of order $p$; equivalently, the number o…

Let $S$ be a finite simple group not isomorphic to $L_2(q)$, where $q$ is a Mersenne prime, and let $p$ be the largest prime divisor of $|S|$. Let $G$ be a finite group generated b…

Vasil'ev–Grechkoseeva conjecture. Suppose that $L$ is one of the following nonabelian simple groups:

Let $L_n(2)$ denote the projective special linear group over the binary field, and let ${\rm Aut}(L_n(2))$ be its automorphism group, for an integer $n\geqslant 2$. A finite group…

Let $L_n(2)$ denote the projective special linear group over the binary field, for an integer $n\geqslant 2$. A finite group is OD-characterizable if it is uniquely determined, up…