26 problems

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 $G$ 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…

Boone--Higman conjecture. Every finitely generated group with solvable word problem embeds in a finitely presented simple group.

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

Lisi–Sabatini conjecture. There exists an element $x \in G$ such that, for each $i$, $P_i \cap P_i^x$ is inclusion-minimal in the set

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

Vdovin's base-two conjecture. One should have

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 $H$ 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

Let $G$ be a finite group, and let $H$ be a finite non-abelian simple group. Denote by the character degrees of a group the degrees of its complex irreducible characters. Huppert's…

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

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…

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

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…

Let $({\mathcal G}_{\mathrm{met}},\|\cdot\|)$ be a metric ultraproduct of metric groups. A metric group is metrically uniformly simple if, for every $r>0$, there is…

Let $G$ be a non-abelian finite simple group, and let $S_1,\dots,S_k$ be normal subsets of $G$. Normal product decomposition conjecture. There exists $c$ such that, whenever … one…

Let $G$ be a non-abelian finite simple group, and let $S_1,\dots,S_k$ be subsets of $G$. Ideal product decomposition conjecture. There exists $c$ such that, whenever … there exist…

Mixed counting conjecture. If

Herzog's conjecture. If $I_2(G)=I_2(H)$, then $|G|=|H|$.

Larsen–Shalev conjecture. For each such word $w$, there exist constants $N(w),\eta(w)>0$ such that, for every nonabelian finite simple group $S$ with $|S|\geq N(w)$,

Let $G$ be a finite group, and let $\mathfrak{p}(G)$ denote its subgroup permutability degree. Subgroup permutability degree criterion. If … then $G$ is not simple. This is present…

Let $G$ be a finite simple classical or alternating group. The probability that two subgroups of $G$ permute is the subgroup permutability degree of $G$, denoted $\mathfrak{p}(G)$.…

Generic-type conjecture. The generic type of $G$ is ESN.