17 problems

Kimmerle's Prime Graph Question. Do the GK-graphs of $G$ and $\operatorname{V}(\mathbb{Z}G)$ coincide?

Mazurov's conjecture. If $G$ is not isomorphic to $A_6$ and $\Gamma(G)$ has at least three connected components, then $G$ is recognizable by its spectrum.

Maslova's conjecture. A prime graph cannot be triangle free if it is not $3$-colorable.

Let $T$ be a non-abelian simple group, and let $G$ be a $T$-solvable group. Write $\operatorname{\overline{\Gamma}}(G)$ for the prime graph complement of $G$. A group is almost qua…

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…

Let $f\geq 5$ be prime, and consider the eight rooted graphs listed in the source, with the first four graphs forming the first row. A graph is realizable by a group if it is isomo…

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

Let $G$ be a finite group with multiple nonisomorphic $K_4$ composition factors, and let its prime graph be the graph whose vertices are the prime divisors of $|G|$, with two disti…

Let $G$ be a finite group, and let $\Delta(cs_{\mathbb{R}}(G))$ be the graph whose vertices are the primes dividing real conjugacy class sizes of $G$, with two distinct vertices ad…

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 $0$-…

Let $G$ be a finite group isospectral to a finite nonabelian simple group $L$. Suppose that $G$ is nonsolvable and has a normal series … where $K$ is the solvable radical of $G$,…

Let $G$ be a finite group. Write $\V(G)$ for the vertex set of the degree graph $\Delta(G)$, and let $\omega(G)$ denote the size of a largest clique in $\Delta(G)$. For each positi…

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…

Keller's conjecture. For every solvable group $G$,