37 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 $G$ be a solvable group, and let $\psi,\varphi\in\operatorname{Irr}(G)$ be faithful, where $\operatorname{Irr}(G)$ denotes the irreducible characters of $G$. Suppose that … whe…

Let $G$ be a quasi-Engel group, meaning that for the fixed initial word $w$ there is an integer $n$ such that the identity ${}^w\!u_n(x,y)\equiv 1$ holds in $G$. A group is residua…

Let $G$ be an infinite finitely generated solvable group, and let $n$ be a square-free integer. Nonexistence conjecture. There are no such groups $G$ satisfying a generalized ident…

For a finite group $G$, let $\operatorname{Irr}(G)$ be the set of irreducible characters and define the set of irreducible character degrees by … Let $\operatorname{dl}(G)$ denote…

Solubility conjecture. Then $G$ is soluble.

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 $G=AB$ be a mutually permutable product of two $p$-solvable subgroups $A$ and $B$, where $p\in\pi(G)$. Write $l_p(G)$ for the $p$-length of $G$, and let $\tau_p(A)$ and…

Let $G$ be a finite group with trivial solvable radical, and let $d_G(H)$ denote the depth of a subgroup $H\leqslant G$. A subgroup is solvable if it is a solvable group. Vdovin's…

Nilpotent-type skew brace conjecture. The group $G$ is isomorphic to the multiplicative group of a skew left brace of nilpotent type.

Solvability conjecture for inverse-class products. If

Let $n$ be a positive integer, and let $(X,\Delta)$ be an $n$-dimensional log canonical log Calabi–Yau pair. Write $\pi_1^{\rm orb}(X,\Delta)$ for its orbifold fundamental group. S…

Let $\{w_j=1\}_{j\in J}$ be a system of equations with variables $\{x_i\}_{i\in I}$ over a group $G$. The system is non-singular when the rows of exponent sums of the variables in…

Let $G$ be a diagram group, and let $\mathbb{Z} \wr \mathbb{Z}^2$ denote the wreath product of $\mathbb{Z}$ by $\mathbb{Z}^2$. Lamplighter exclusion conjecture. The group $G$ does…

Let $G$ be a finite group, let ${\mathbf F}(G)$ denote its Fitting subgroup, and let $\operatorname{Irr}(N)$ be the set of irreducible characters of a normal subgroup $N$. For…

Let $G$ be a finite group, and let $\mathbf F(G)$ denote its Fitting subgroup. Write $\operatorname{Irr}(G)$ for the set of irreducible characters of $G$. Four-character Fitting-in…

Let $G$ be a finite group, and let $\rho(G)$ be the set of primes dividing the degree of some irreducible character of $G$. Write $\operatorname{Irr}(G)$ for the set of irreducible…

Quadratic Schreier growth gap conjecture. $G$ has a Schreier growth gap $n^2$.

Scaling-group conjecture. The scaling group of $G$ is a subgroup, possibly trivial, of

Solvable-group supremum conjecture. For every positive integer $n$, one has

Rational growth conjecture. Every torus bundle group $G=\mathbb{Z}^2\rtimes_A\mathbb{Z}$ has rational growth with respect to some finite generating set.

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.

The average-order solvability conjecture. If

Let $G$ be a solvable group and let $A$ act on $G$ by automorphisms with $(|A|,|G|)=1$. Let $F(G)$ be the Fitting subgroup, define $F_0(G)=1$, $F_1(G)=F(G)$, and let $F_{i+1}(G)$ b…

Let $G$ be a finite group, and let $v_p(G)$ denote the number of Sylow $p$-subgroups of $G$. Robinson's solvability conjecture. If … for each odd prime number $p$, then $G$ is solv…