40 problems

Let $G$ be a nonabelian finite simple group, and let $\operatorname{diam}(G)$ denote the maximum of $\operatorname{diam}(G,X)$ over all generating sets $X$ of $G$. Babai’s conjectu…

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…

Vdovin's conjecture. Then $H \cap H^x = 1$ for some $x \in G$.

Lisi–Sabatini conjecture for simple groups. There exists an element $x \in G$ such that

Let a group $\Gamma$ act faithfully on a set $S$. The action is of type (A) if $\Gamma$ is finitely presented, every stabilizer $\operatorname{Stab}_{\Gamma}(s)$ is finitely genera…

Let $G$ be a finite simple group, and let $H$ be a subgroup of $G$ whose index $[G:H]$ is odd. Vdovin–Revin conjecture. The subgroup $H$ is pronormal in $G$. This conjecture concer…

Let $\operatorname{Homeo}_{+}(\mathbb{S}^{n})$ denote the group of orientation-preserving homeomorphisms of the sphere $\mathbb{S}^{n}$. Simplicity conjecture. A simple application…

Let $\Lambda$ be the finitely presented $(6,6)$-group defined by the generators and relators in the preceding construction, and let $\Lambda_0<\Lambda$ be the subgroup under consid…

Let $\overrightarrow{u}$ be an autocorrect sequence, and let a $\overrightarrow{u}$-Engel automorphism mean an automorphism satisfying the corresponding Engel condition. Let $G$ be…

Let $S$ be the finitely generated simple group constructed in the paper, whose word problem is coNP-complete. It is explicitly embedded in the Thompson–Higman group…

Let $\mathcal{L}$ be the list of nonabelian simple groups not yet covered by the general recognition results: $L_n(q)$ with $8\leq n\leq26$ and $n$ not prime; $U_n(q)$ with…

Vdovin's base-two conjecture. One should have

Let $A$ be a countable abelian group, and write $\bigoplus_\omega(\mathbb{Q}\oplus\mathbb{Q}/\mathbb{Z})$ for the countable direct sum of copies of…

Let $H$ be a finite nonabelian simple group, and let $G$ be a finite group. Write ${\mathrm {cod}}(G)$ for the set of character codegrees of $G$. Codegrees conjecture. If … then…

The codegree isomorphism conjecture. Let $H$ be a finite nonabelian simple group and $G$ a finite group. Then

Burness–Guralnick conjecture. $G$ can be generated by a Sylow $r$-subgroup and a Sylow $s$-subgroup.

Let $l\in\mathbb{N}$. A non-singular mixed identity for a group $G$ is a non-trivial word in the free product $G\ast\mathbf{F}_r$ that is satisfied under the relevant substitutions…

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

Blackburn's conjecture. As the order of $S$ tends to infinity,

Let $G$ be a finite simple group, and let $r$ and $s$ be any prime divisors of $|G|$, not necessarily distinct. A Sylow $r$-subgroup and a Sylow $s$-subgroup of $G$ are subgroups a…

Let $G$ be a finite non-abelian simple group, let $t\in G$ be an involution, and let $\mathcal{Z}_G(t)$ denote the centralizer of $t$. A coset of the centralizer is a set…

Let $G$ be a finite group. An element $g\in G$ is a coprime commutator if $g=[x,y]$ for elements $x,y\in G$ whose orders are coprime. Pellegrini–Shumyatsky's conjecture. Every elem…

Cross-characteristic composition-factor conjecture. The group $S$ cannot be a group of Lie type in characteristic coprime to $q$.

Conjecture. If $K$ is the solvable radical of $G$, then

Let $G$ be a finite simple group, and let $r$ and $s$ be primes dividing $|G|$. A Sylow-subgroup generation conjecture. There exist a Sylow $r$-subgroup $P$ and a Sylow $s$-subgrou…