71 problems

Thompson's conjecture. Every finite nonabelian simple group $S$ is uniquely determined by the set $\operatorname{cs}(S)$ in the class of all finite groups with trivial center; equi…

Arad–Herzog conjecture. The product of two nontrivial conjugacy classes of $G$ is never a conjugacy class.

Solvability conjecture. The subgroup $\langle K\rangle$ is solvable.

Primitive character divisibility conjecture. The character degree $\chi(1)$ divides the size of some conjugacy class:

Aschbacher–Guralnick conjecture. One has

Solubility conjecture. Then $G$ is soluble.

Let $G$ be a finite nonabelian simple group, and let $K$ and $L$ be nontrivial conjugacy classes of $G$. Arad–Stavi–Herzog conjecture. The product $KL$ cannot be a single conjugacy…

Herzog–Kaplan–Lev conjecture. One has

Let $G$ be a finite solvable group and let $A$ be a conjugacy class of $G$. Write $\eta(AA^{-1})$ for the number of conjugacy classes occurring in the product $AA^{-1}$, and let…

Let $\kappa=2$, and consider tuples of conjugacy classes for which conditions i) and ii) of the source's genericity theorem hold, but for which the weak Deligne-Simpson problem is…

Ore--Thompson conjecture. For every finite simple group $G$, $\operatorname{or}(G)=2$. The conjecture concerns products of elements from a single conjugacy class and is used in the…

Exponential conjugacy-growth conjecture. The conjugacy growth of $G$ is exponential.

Let $G$ be a finite group. Let $f(G)$ denote the number of irreducible characters of $G$ with rational fields of values, and let $h(G)$ denote the number of conjugacy classes of…

Let $G$ be a finite group. Write $\operatorname{Irr}_{\mathbb{Q}}(G)$ for the irreducible characters of $G$ whose fields of values are contained in $\mathbb{Q}$, and write…

Szép's conjecture. No non-abelian finite simple group $G$ has a factorisation of the form

Let $G$ be a finite group, let $K$ be a conjugacy class of $G$, and suppose that … for some conjugacy class $D$ of $G$. Beltrán–Felipe–Melchor conjecture. The subgroup generated by…

Let $G=\operatorname{Inn}(G)$ be a non-abelian finite simple group, and let $\alpha\in\operatorname{Aut}(G)$. Coset-square conjecture. There exists $x\in\alpha G$ such that, for th…

Let $G$ be a finite group, let $C$ and $D$ be conjugacy classes of $G$, and let $n\geq 2$. Beltrán et al.'s conjecture. If … then $\gen{C}$ should be soluble. This conjecture propo…

Let $G$ be a group, let $X$ be a subset of $G$, and let $H:=\langle X\rangle\leq G$. Assume that $X$ is a conjugacy class of $H$. Central-translate conjecture. If there exist disti…

For a nonnegative integer $n$, let $\mathcal{D}_n$ be the family of finite groups with exactly $n$ conjugacy classes of nontrivial, non-self-normalizing subgroups. For a finite gro…

Let $n\in\mathbb{N}^{*}$, let $q$ be a prime power, and let $U_n(\mathbb{F}_q)$ be the group of $n\times n$ upper triangular matrices over $\mathbb{F}_q$ with ones on the diagonal.…

Let $G$ be a group, and let $p$ be a prime. An element of $G$ is p-regular if its order is not divisible by $p$, and the conjugacy class size of an element $x$ is $|G:C_G(x)|$. The…

Let $G$ be a finite non-abelian simple group, and let $A$ and $B$ be two non-trivial conjugacy classes of $G$. Arad–Fisman conjecture. The product $AB$ is not a conjugacy class of…

Sharp $p-1$ factorization conjecture. One should have

Arad–Hergoz conjecture. The set $AB$ cannot be a single conjugacy class of $S$.