66 problems

Let $G$ be a finite simple group and let $A\subseteq G$ be a subset with $|A|\geq 2$. A conjugate of $A$ means a set $gAg^{-1}$ for some $g\in G$. Liebeck–Nikolov–Shalev conjecture…

An unmixed Beauville structure is a pair of generating triples satisfying the required Beauville disjointness condition. Beauville's conjecture. Every finite non-abelian simple gro…

Spiga's element-and-involution conjecture. Except for $\mathrm{PSL}_2(q)$ and a finite number of other cases, every finite non-abelian simple group $G$ contains an element $x$ and…

Babai–Seress conjecture. There is an absolute constant $c$ such that the Cayley diameter of every finite non-abelian simple group $G$ is at most

Let $G$ be a finite simple group and let $H\leq G$ be a subgroup of odd index. A subgroup $H$ is pronormal in $G$ if, for every $g\in G$, $H$ and $H^g$ are conjugate in…

Let $w_1$ and $w_2$ be non-trivial group words, and for a group $G$ let $w_i(G)$ denote the set of values of $w_i$ in $G$. Two-word Waring conjecture. There exists an integer…

Magnus's conjecture. The free group $F_{2}$ is residually $S$ for every such set $S$.

Let $S$ be a simple group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $S$. A group element is a $p$-element if its order is a power of $p$. Conder's conjecture. If…

Chen's conjecture. There do not exist any $(G,3)$-arc-transitive, $G$-vertex-primitive digraphs with $G$ almost simple.

Let $F_2$ be the free group on two generators. A group is fully residually finite simple if, for every finite subset of its nontrivial elements, there is a homomorphism to a finite…

Let $F_n$ denote the free group on $n$ generators. A subgroup $C\leq F_n$ is characteristic if it is preserved by every automorphism of $F_n$. The Baby Wiegold conjecture. For ever…

For a prime $p$, let $O_p(G)$ be the intersection of all Sylow $p$-subgroups of $G$. Say that $G$ has $(*)_p$ if there exist two Sylow $p$-subgroups whose intersection is contained…

Let $G$ be a finite group, and let $S$ be a non-abelian simple group. Write $\mu(G)$ for the set of proportions of $r$-singular elements of $G$ as $r$ ranges over the primes dividi…

Let $p$ be a prime and let $S$ be a finite non-abelian simple group whose order is divisible by $p$. Simple-group reduction conjecture. The following assertions hold. (a) Let $X$ b…

Almost-all non-congruence presentations conjecture. As the order of $S$ tends to infinity, the proportion of non-congruence presentations among all presentations tends to $1$:

Let $\Sigma$ be the infinite family of finite non-abelian simple groups … Suppose that a profinite group $G$ satisfies $P_2(G)>0$, where $P_2(G)$ denotes the proportion of 2-elemen…

Cocycle-Cheeger conjecture. The cocycle Cheeger constants in permutations of the Steinberg presentations have a uniform positive lower bound, at least when $n$ is fixed and…

Abdollahi–Akbari–Maimani conjecture. If

Prime-parameter K4 conjecture. There are infinitely many primes $r$ such that

Vo-set characterization conjecture. $G\cong S$ if and only if

Two-orders characterization conjecture. $G\cong S$ if and only if

A graphical regular representation (GRR) of a group $G$ is a Cayley graph whose full automorphism group is isomorphic to $G$; a GRR is $k$-valent when the corresponding graph has v…

Let $p$ be a prime and let $q$ be an odd prime power with $p\mid q+1$. Consider the $(\mathcal{Q}\mathcal{D})$-List of possible $p$-extensions of finite simple groups that may fail…

Let $c$ be a positive integer, and let $G$ be a finite non-abelian simple group. Write $\gamma(G)$ for its normal covering number. Bounded normal covering conjecture. There exists…

Let $G$ be a finite non-abelian simple group. Write $\gamma(G)$ for its normal covering number and $\gamma_w(G)$ for its weak normal covering number, and let $m\in\mathbb{N}$. Weak…