9 problems

Amenability conjecture. A connected Lie group $G$ has $m(G)<\infty$ if and only if $G$ is amenable.

Gelander's conjecture. For every connected compact simple Lie group $G$ and every connected simple $\mathbb{C}$-algebraic group $\mathbf{G}$,

For integers $n$ and $k$ with $n>k+1$, define … Maximum-row-size conjecture. … The conjecture is motivated by computational experiments and the displayed generating-triple patterns…

Let $n,m,k$ be integers with $2\leqslant k<m<n$. Mod-2 persistence conjecture. For any $n\equiv0,1\pmod{4}$, if … then … This was proposed from computational experiments for…

For $n$ and integers $m,k$ with $2\leqslant k<m<n$, let $f(n)$ be the number of pairs $(m,k)$ for which the prefix reversals $r_n,r_m,r_k$ generate the symmetric group…

Let $g_3^{\ast}(n)$, $g_3^{\ast\ast}(n)$, and $g_r(n)$ be the functions defined earlier in the paper, and let $\mathbb{N}^{\geq k}=\{n\in\mathbb{N}:n\geq k\}$. Equality and strict…

Let $G$ be a finite group, and let $G_p$ be a Sylow $p$-subgroup of $G$ for each prime $p$. Define … where $d(G_p)$ is the minimum size of a generating set of $G_p$, and let $m(G)$…

Let $G=\mathrm{PSL}(2,p)$), and let $\iota_n(G)$ denote the set of divisors occurring as the relevant invariant for irredundant generating sequences of length $n$ in the paper. Fou…

Let $G=\mathrm{PSL}(2,p)$, where $p$ is prime. An irredundant generating set is a generating set such that removing any one generator leaves a collection that no longer generates…