16 problems

Let $G$ be a finite group and let $P\in\operatorname{Syl}_2(G)$. Let $\sigma\in\operatorname{Gal}(\mathbb{Q}^{\mathrm{ab}}/\mathbb{Q})$ be the unique automorphism that fixes $2$-ro…

Lisi–Sabatini conjecture for simple groups. There exists an element $x \in G$ such that

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Write ${\mathrm {cd}}(P)$ for the set of irreducible character degrees of $P$, and…

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 $(P_i)_{i=1}^n$ be Sylow subgroups for distinct primes $p_1,\ldots,p_n$. Synchronization conjecture. There exists $x \in G$ such that, for all…

Fix a prime $p$. Let $G\unlhd A$ be finite groups, let $P$ be a Sylow $p$-subgroup of $G$, and write $\operatorname{IBr}(H)$ for the set of $p$-Brauer characters of a group $H$. Sy…

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Write $\operatorname{IBr}(G)$ for the set of $p$-Brauer characters of $G$. Malle–Navarro–…

Let $p$ be an odd prime, and let $G$ be a finite group with subgroup $H$. For a finite group $X$, write $up(X)$ for the number of Sylow $p$-subgroups of $X$. **Gap conjecture for…

Navarro–Tiep conjecture. The following conditions are equivalent: $d_{\chi 1_G}\neq 0$ for every $\chi\in{\operatorname{Irr}}_{p'}(G)$; $G$ has self-normalising Sylow $p$-subgroups…

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Let $\zent P$ denote the center of $P$, and for an irreducible character $\chi$ of $G$, l…

Let $G$ be a finite simple group, and let $r$ and $s$ be any prime divisors of $|G|$, not necessarily distinct. A Sylow $r$-subgroup and a Sylow $s$-subgroup of $G$ are subgroups a…

Let $G$ be a finite group, let $T$ be a Sylow-$2$ subgroup of $G$, and let $t\in Z(T)$ be an involution. Let $\mathcal{Z}_G(t)$ denote the centralizer of $t$. Pellegrini–Shumyatsky…

Let $G$ be a finite group, and let $v_p(G)$ denote the number of Sylow $p$-subgroups of $G$. Robinson's solvability conjecture. If … for each odd prime number $p$, then $G$ is solv…

Let $G$ be a finite group and let $P$ be a Sylow $2$-subgroup of $G$, with $4\mid |G|$. Let $B_0(G)$ be the principal $2$-block, and let…

Let $G$ be a finite group and let $P\in\operatorname{Syl}_2(G)$. Let $\sigma\in\operatorname{Gal}(\mathbb{Q}^{\mathrm{ab}}/\mathbb{Q})$ be the unique automorphism that fixes $2$-ro…

Almost simple group conjecture. If $S$ satisfies $\mathbf{DivSyl}(p)$, then every group $L$ satisfying