9 problems

Let $p$ and $q$ be prime powers such that $\operatorname{PSL}(2,q)$ and $\operatorname{PSL}(2,p)$ are $K_4$-groups. Suppose one of the following holds: $p$ and $q$ are primes with…

Prime-graph complement characterization. For every strictly $T$-solvable group $G$, there exists a set $Y\subseteq\bigcup_{p\in\pi(T)}\operatorname{IBr}_p(T)$ such that

Super-base graph conjecture. For every such $n$ and $k$, if $G_{n,k}$ is a minimal prime graph, then $\overline{G_{n,k}}\setminus v$ is not a minimal prime graph for any vertex $v$…

A prime graph is a graph with no nontrivial module, and an infinite prime graph is a prime graph with infinitely many vertices. Let $G_\mu$ denote the graph associated with a $0$-…

Maslova's conjecture. If the prime graph of $G$ does not contain 3-cocliques, then it is isomorphic to the Gruenberg–Kegel graph of some finite solvable group.

Let $n\geqslant 12$ be an even integer, and let $p$ be the largest prime less than $n$. Distinct-prime Goldbach variation. There exist distinct primes $r$ and $s$ such that … This…

Let $G=A_n$ and $H=(S_k\times S_{n-k})\cap G$, where $1<k<n$ and $p\leqslant k$ for every prime number $p\leqslant n$. Prime-graph equality conjecture. If $n\geqslant 12$, then … i…

Let $G$ be a finite group, let $\rho(G)$ denote the set of primes dividing character degrees of $G$, and let $\Delta(G)$ be the prime graph of $G$, whose vertices are the elements…

Let $G$ be a finite group. Write $\#(G)$ for the set of primes dividing the order of $G$. The Gruenberg–Kegel graph, or prime graph, of $G$ is the graph $\pi(G)$ whose vertices are…