30 problems

Let $G$ be a finite group. A proper quotient means a quotient $G/N$ with $1\neq N\trianglelefteqslant G$, and $s(G)$ denotes the largest integer $k$ such that any $k$ nontrivial el…

Three-involution generation conjecture. The upper bound of four in the stated generation results is not sharp: $\operatorname{Map}(S(n))$ is generated by three involutions for all…

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…

Hamiltonian generating-graph conjecture. The generating graph $\Gamma(G)$ is Hamiltonian if and only if every proper quotient of $G$ is cyclic.

Maximal-subgroup generating density conjecture. For every finite group $G$,

One-outside-element conjecture. There exists a generating set of $G$ of size $d(G)$ having only one element outside $M$.

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

Let $F$ be Thompson's group, and let a finite index subgroup mean a subgroup $H\leq F$ with finite index $[F:H]$. The finite-index subgroup 2-generation conjecture. Every finite in…

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

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 $G$ be a finite group, let $d(G)$ be the smallest size of a generating set for $G$, and let $k\geqslant d(G)$ be an integer. The product replacement graph $\Gamma_k(G)$ has as…

Strong $\frac{3}{2}$-generation conjecture. One should have

Let $G$ be a finite group, let $m(G)$ be the largest size of a minimal generating set of $G$, and let … where $\pi(G)$ is the set of prime divisors of $|G|$ and $d_p(G)$ is the min…

Dennis' conjecture. For every finite group $G$,

Let $G_i$ be a sequence of finite simple groups with $|G_i|\to\infty$, and suppose that the primes $r$ and $s$ both divide $|G_i|$. Let $P_{r,s}(G_i)$ denote the probability that a…

For primes $r,s$, let $P_{r,s}(G)$ be the probability that a uniformly random pair of elements of orders $r$ and $s$ generates the finite simple group $G$, and let $L_{r,s}$ be the…

Let ${\widehat\Gamma}_{X_6}$ be the saturated cluster modular group of type $X_6$, and let $\Gamma_{X_6}$ be the cluster modular group of type $X_6$. A cluster Dehn twist is an inf…

Let $G$ be an almost simple group, meaning that $T\leq G\leq {\rm Aut}(T)$ for some non-abelian finite simple group $T$, and let $H$ be a maximal subgroup of $G$. Write $d(H)$ for…

Let $G$ be a finite group with $|G|\geq 4$, and let $\Gamma(G)$ be its generating graph. Equivalence conjecture. The following conditions are equivalent: $G$ has spread $1$; $G$ ha…

For a finite group $G$, being $(2,r)$-generated means that $G$ is generated by an involution and an element of order $r$. Conder's conjecture. Every non-abelian finite simple group…

Two-generator conjecture. Two elements of order $k$ suffice to generate $\Sigma_n$ when $k$ is even and to generate $\operatorname{A}_n$ when $k$ is odd.

Let $G$ be a finite group with $|G|\geqslant 4$, and let $\Gamma(G)$ be its generating graph. The spread of $G$ is the largest integer $k$ such that, for every set of $k$ nonidenti…

A profinite group $G$ is positively finitely generated (PFG) if, for some finite $k$, a random $k$-tuple of elements generates $G$ with probability $P(G,k)>0$. Define the Dirichlet…

A generalised Wilson group is an infinitely iterated wreath product of finite non-abelian simple permutation groups of the type considered in the paper. Its lower rank is the least…

Let $G$ be a finite simple group, and let Condition denote the prime-generation condition introduced earlier in the paper. Dolfi–Guralnick–Herzog–Praeger conjecture. The analogue o…