6 problems

Thompson's conjecture. $G \cong S$ if and only if $|G|=|S|$ and $\pi_e(G)=\pi_e(S)$.

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 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…

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…