6 problems

Commutator Baer–Suzuki conjecture. If $[x,g]$ is either $1$ or $p$-singular for every element $g\in G$, then $x\in O_p(G)$.

Multicommutator Baer–Suzuki conjecture. If, for some integer $k\geqslant 1$, $ab$ is a $p$-element for all $a,b\in\Gamma_k(x)$, then $x\in O_p(G)$.

Guralnick–Malle's conjecture. Then $C\subseteq O_p(G)$.

Let $r$ be a prime, let $L$ be a nonabelian simple group whose order is divisible by $r$, and let $x\in\operatorname{Aut}(L)$ be an automorphism of prime order. Bound conjecture fo…

Let $\pi$ be a proper subset of the set of all primes containing at least two elements, and let $r$ be the minimal prime not in $\pi$. The invariant $\operatorname{BS}(\pi)$ is def…

Let $G$ be a finite group, let $p\ne 5$ be a prime, and let $C$ be a conjugacy class of $p$-elements in $G$. Commutator Baer–Suzuki conjecture. If $[c,d]$ is a $p$-element for all…