11 problems

For every finite group $G$, there exists a finite lattice $L$ such that $\operatorname{Aut}(L)\cong G$ and the action of $\operatorname{Aut}(L)$ on the elements of $L$ has at most…

Determine, up to isomorphism, all simple $5$-$(v,k,\lambda)$ designs with $\lambda=2$ that admit a group of automorphisms acting transitively on their blocks; equivalently, classif…

For every finite group $G$, every prime $p$ dividing $|G|$, and every Sylow $p$-subgroup $P$ of $G$, one has $|\operatorname{Irr}_{p'}(G)|=|\operatorname{Irr}_{p'}(N_G(P))|$, where…

For every finite group $G$, every prime $p$, and every Sylow $p$-subgroup $P\in\operatorname{Syl}_p(G)$, let $\operatorname{Irr}_p(G)=\{\chi\in\operatorname{Irr}(G):p\mid\chi(1)\}$…

Let $G$ and $A$ be finite groups such that $A$ acts on $G$ by automorphisms and $\gcd(|A|,|G|)=1$. For every prime $p$, the number of $A$-invariant irreducible $p$-Brauer character…

Let $G$ be a finite primitive permutation group on a set $\Omega$, and let $b$ be its base size. Define the generalised Saxl graph $\Sigma_b(G)$ to have vertex set $\Omega$, with d…

For every finite solvable group $G$, every element $g\in G$ such that $\chi(g)\neq 0$ for every irreducible complex character $\chi\in\operatorname{Irr}(G)$ belongs to the Fitting…

Let $\mathcal{V}_n$ and $\mathcal{D}_n$ denote, respectively, the sets of labelled vertex-transitive graphs and vertex-transitive digraphs on $[n]=\{1,\ldots,n\}$, and let…

For every prime $p$, every finite group $G$, every $p$-block $B$ of $G$, and every defect group $D$ of $B$, one has $\operatorname{mh}(B)\leq \operatorname{mh}(D)$, where…

For every finite simple nonabelian group $G$, there exists a conjugacy class $C$ of $G$ such that $C^{2}=\{xy:x,y\in C\}=G$.

Godara and Sarkar conjectured that, for every odd prime $p$, if $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$ is the Heisenberg group of order $p^3$, then $\mathsf{d}(H_{p^3})=3p-3$,…