13 problems

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…

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…

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 $X$ be a finite almost simple group, meaning that its socle $\operatorname{soc} X$ is a nonabelian simple group and…

Wiegold's conjecture. If $r\geq 3$, then $\operatorname{Aut}(\mathbf{F}_{r})$ acts transitively on $\operatorname{Epi}(\mathbf{F}_{r},G)$. This is equivalent to asking whether all…

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…

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…

Prime-order generation conjecture. With finitely many exceptions, there exist conjugacy classes $C$ and $D$ of $G$ consisting of elements of prime order such that

Let $G$ be a finite nonabelian simple group. Define … to be the set of element orders of $G$. Prime-order generation conjecture. There exist two distinct primes…

Let $\mathbb{K}$ be a field and let $n$ denote the number of variables. A polynomial automorphism is linearizable if it can be transformed into a linear automorphism by a polynomia…

Let $\mathbb{K}$ be the field appearing in the polynomial automorphism group $\operatorname{GA}_3(\mathbb{K})$. An automorphism fixing one variable is a polynomial automorphism tha…