14 problems

Neumann–Praeger conjecture. Is there a function $f:\mathbb{N}\to\mathbb{N}$ such that whenever $U<H\leq A$, where $H$ is a normal subgroup of the finite group $A$ of index $n$ and…

Covering conjecture. The set $G_p$ cannot be covered by $p$ proper subgroups of $G$.

Let $A$ be a finite group with normal subgroup $G$ of index $|A:G|=n$, and let $U\leqslant G$ satisfy $P_A(U)=P_A(G)$, where $P_A(V)$ is the union of the $A$-conjugacy classes of e…

For a finite group $A$ and a subgroup $U\leqslant A$, let $\mathcal{P}(U)$ be the set of elements of $U$ of prime-power order, and define … the union of the $A$-conjugacy classes o…

A5 composition-factor conjecture. If

PSL2 congruence conjecture.

Characteristic-two PSL2 conjecture.

Neumann–Praeger covering conjecture. There is an integer function $g$ such that

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…

Let $G$ be a finite simple group. An equal covering of $G$ is a covering by proper subgroups whose members all have the same order. Equal-covering conjecture. If $G$ is a finite si…

For a group $G$, let its covering number $\sigma(G)$ be the minimum number of proper subgroups whose union is $G$, with $\sigma(G)=\infty$ if no finite covering exists. Covering-nu…

Let $\Sigma$ be a property of profinite groups such that every finite group is a $\Sigma$-group, the class of $\Sigma$-groups is closed under taking subgroups, quotients and extens…

Let $G$ be a profinite group whose commutator subgroup is finite-by-procyclic. For a subgroup $K$, write $\pi(K)$ for the set of primes dividing the orders of elements of $K$. Fini…