4 problems

Let $\pi$ be a fixed subset of the set $\mathbb{P}$ of all primes with $\pi'=\mathbb{P}\setminus\pi\ne\varnothing$. A $\pi$-group is a finite group whose order has all prime diviso…

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

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…