34 problems

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…

Solubility conjecture. Then $G$ is soluble.

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 $G=\operatorname{Inn}(G)$ 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 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…

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…

Solvability conjecture for inverse-class products. If

Arad–Herzog conjecture. This product is not a conjugacy class.

Let $p$ be a prime. Let $a$ and $b$ be positive integers such that … and such that $|a-b|$ is minimal. For a finite group $G$, let $k(G)$ denote the number of conjugacy classes of…

Let $G$ be a finite group, let $k(G)$ denote its number of conjugacy classes, and let $|G|$ denote its order. Linear conjugacy-class bound. There exists a constant $C>0$ such that,…

Non-$p$-solvable conjugacy-class conjecture. If $G$ is non-$p$-solvable, then

Generalized exponent conjecture. The generalized exponent of $G$ is at most $3$.

Herzog–Kaplan–Lev conjecture. One has

Let $\Gamma_g$ be the Cayley graph of a closed surface group with respect to a standard generating set. For a conjugacy class $\mathfrak{c}$, let $\lVert\mathfrak{c}\rVert$ denote…

Let $n(k,l)$ be the largest integer $n$ such that every element of the alternating group $A_n$ is a product of $k$ many $l$-cycles, where $k,l\geq 2$ and either $l$ is odd or $k$ i…

Let $L$ be the ambient subgroup, let $h(t)$ count the $\operatorname{SL}_2(\mathbb{Z})$-conjugacy classes of hyperbolic matrices in $L$ with trace $t$, and let $\ell([g])$ denote t…

Let $L$ be the ambient subgroup under consideration, and let $h(t)$ be the number of $\operatorname{SL}_2(\mathbb{Z})$-conjugacy classes of hyperbolic matrices in $L$ with trace…

Let $A$ be an almost simple primitive group on $m$ points appearing in the paper's table of final exceptions, and assume that $k(A)<m$. For each primitive subgroup…

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

Conjugacy-ratio conjecture. The group $G$ has positive conjugacy ratio if and only if $G$ is virtually abelian:

Let $k$ be an odd number. A finite $2$-group is 2-generated if it can be generated by two elements, and a real conjugacy class is a conjugacy class consisting of elements conjugate…

Let $r$ be an odd natural number. A finite group element is real if it is conjugate to its inverse, and a conjugacy class is real if all its elements are real. Sangroniz's conjectu…

For $n\geq 1$, let $U_n(q)$ be the group of upper triangular $n\times n$ matrices over $\mathbb F_q$ with ones on the diagonal, and let $k(U_n(q))$ denote its number of conjugacy c…