46 problems

Let $psi_k$ denote the staircase (2-defect zero) irreducible characters under discussion, and let a character have the tensor-square property when every irreducible character occur…

Let $\Gamma$ be a Fuchsian group, namely a finitely generated non-elementary discrete group of isometries of the hyperbolic plane. Higman's conjecture. Every Fuchsian group surject…

Herzog–Kaplan–Lev conjecture. One has

For a finite group, consider the set of dimensions of its irreducible complex representations. In particular, consider the alternating group $A_n$ and the symmetric group $S_n$. To…

Let $H$ be a finite group, and let $h_{\rm OD}(H)$ denote the number of isomorphism classes of groups $G$ such that $|G|=|H|$ and ${\rm D}(G)={\rm D}(H)$. A group is OD-characteriz…

Let $A_n$ be an alternating group, and call a subset $X\subseteq A_n$ product-free if there are no $x,y,z\in X$ with $xy=z$. A maximal subgroup is a proper subgroup maximal under i…

Alternating-group Beauville structure conjecture. Let $(r,s,t)$ and $(r',s',t')$ be two hyperbolic types. Then almost all alternating groups ${\mathcal{A}}_n$ have an unmixed Beauv…

Call $(r,s,t)\in{\mathbb N}^3$ hyperbolic when … For an unmixed Beauville structure $v=(a_1,c_1;a_2,c_2)$, its two generating systems have types $(r,s,t)$ and $(r',s',t')$, respect…

Let $A_n$ denote the alternating group of degree $n$. A self-dual string C-group is a string C-group representation whose associated polytope is self-dual. Existence conjecture. Fo…

Let $A_n$ be an alternating group. A complex irreducible character $chi$ of $A_n$ has the tensor-square property if every irreducible character of $A_n$ occurs as an irreducible co…

Let $F_2$ be the free group on two generators, and let $A_n$ denote the alternating group on $n$ elements. A finite quotient of $F_2$ is characteristic when it is of the form…

Let $C_n$ be the cactus group acting on the set of standard Young tableaux of a partition, and let $A_N$ denote the alternating group on $N$ tableaux, where $N$ is the number of st…

Let $A$ be the union of the alternating subgroups, let $S_n$ be the symmetric group on $n$ letters, and let $P$ be a basis of mesh patterns such that $A=\operatorname{Av}(P)$. A me…

Alternating-group derangement bound conjecture. The following hold:

Alternating-group derangement-class square conjecture. In this case,

The alternating-group derangement formula. The displayed formula holds for every $n\geqslant 9$.

Let $S$ be a non-abelian finite simple group, and let $A,B$ be subsets of $S$. For $g\in S$, write $A^g=g^{-1}Ag$. Asymmetric growth conjecture. For every $\delta>0$ there exists…

Fulman–Kim–Lee–Petersen conjecture. The polynomials $\sum_{i=0}^n P_{n,k}t^k$ and $\sum_{i=0}^n Q_{n,k}t^k$ are all real-rooted for $n\ge 5$. This is an open problem proposed as pa…

Let $G=A_n$ be an alternating group, and let $A$, $B$, and $C$ be normal subsets of $G$. For a subset $X\subseteq G$, write $\mu(X)=|X|/|G|$ for its density. Garonzi–Maróti conject…

Let $A_n$ be the alternating group, and let the density of a subset of $A_n$ be its size divided by $|A_n|$. A subset is normal if it is a union of conjugacy classes. For subsets…

Let $A_n$ be the alternating group. A subset $A\subseteq A_n$ is normal if it is a union of conjugacy classes, and its density is $|A|/|A_n|$. For a subset $A$, write…

Let $n(k,l)$ be the largest integer $n$ such that every element of the alternating group $A_n$ is a product of $k$ many $l$-cycles, where $k,l\geq 2$ and either $l$ is odd or $k$ i…

Let $A_n$ be the alternating group, and call a subset $F\subseteq A_n$ product-free if there are no $a,b,c\in F$ with $a=bc$. For $x\in\{1,\ldots,n\}$ and…

Guralnick–Shareshian–Woodroofe's question. Is $A_n$ invariably generated by an element of order $p^a$ together with an element of order $r$?

Let $G$ be a finite non-abelian simple group, let $t\in G$ be an involution, and let $\mathcal{Z}_G(t)$ denote the centralizer of $t$. A coset of the centralizer is a set…