95 problems

Let $G$ be a finite group, and let $\mathbf F(G)$ denote its Fitting subgroup. Let $b(G)$ be the largest degree of the irreducible characters of $G$. Gluck's conjecture. If $G$ is…

Fitting-height conjecture. If $A$ acts fixed-point-freely on $G$, then

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…

Almost all almost simple groups conjecture. Almost all almost simple groups are minimal wexp-nonsolvable.

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

Isaacs–Navarro–Wolf's conjecture. If $G$ is solvable, then

Let $G$ be a finite group. For each pair of primes $p$ and $q$, suppose that $G$ contains a Hall subgroup for the set of primes $p,q\}$. P. Hall's solvability conjecture. Then $G$…

Let $G$ be a finite group. The MGSE solvability conjecture. If $G$ satisfies the MGSE property, then $G$ is solvable. The MGSE property concerns the behavior of minimal generating…

Let $M,N\in\operatorname{SL}(n,\mathbf{Z})$ lie on $1$-parameter subgroups of $\operatorname{GL}(n,\mathbf{R})$, and suppose that neither $M$ nor $N$ has an eigenvalue on the unit…

Solubility conjecture. Then $G$ is soluble.

Fitting-subgroup conjecture. Every nonvanishing element $x$ of $G$ lies in $\mathbf{F}(G)$.

A skew brace is a set with two group structures, an additive group $(A,+)$ and a multiplicative group $(A,\circ)$, satisfying the skew brace compatibility law. A skew brace is of s…

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 Lie group, and let $\Gamma$ be a lattice in $G$. A group is virtually a lattice if it is weakly commensurable to a lattice, where weak commensurability is gen…

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…

Solvable quotient conjecture. Every non-adorable finitely presented torsion free group has an infinite solvable quotient.

Moishezon–Teicher's conjecture. For many classes of simply-connected algebraic surfaces, the groups $\pi_1(\mathbb{CP}^2-S_V)$ will be extensions of symmetric groups by solvable gr…

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…

Solvability conjecture. If

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…

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, let $p$ be a prime, and let $x\in G$ be a $p$-element. Suppose that for every $g\in G$, the commutator $[x,g]$ is either $1$ or $p$-singular, meaning tha…