22 problems

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…

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…

Alternating-group derangement bound conjecture. The following hold:

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…

Herzog–Kaplan–Lev conjecture. One has

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 $G$ 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…

Let $p\geqslant 11$ be a prime with $p\equiv 3\pmod{4}$. For the alternating group $A_p$, let $\delta_{\mathcal{S}}(A_p)$ denote the diameter of its soluble graph. Alternating-grou…

Exponential graph-size conjecture. There exists a constant $C>1$ such that, if $k\geq3$ and $\operatorname{Aut}(X)$ has an epimorphism onto $A_k$, then

Descent–excedance synchronisation conjecture. For every positive integer $n$, the sequences $(A_{n,k}^+)_{k=0}^{n-1}$ and $(P_{n,k})_{k=0}^{n-1}$ are strongly synchronised. Similar…

Strong-synchronisation conjecture. For every positive integer $n$, the sequences $(A_{n,k}^+)_{k=0}^{n-1}$ and $(A_{n,k}^-)_{k=0}^{n-1}$ are strongly synchronised.

Let $A$ and $B$ be finite groups, let $C$ be a subgroup of $A\cap B$ of index at least $2$ in $A$ and at least $3$ in $B$, and let $K$ be the core of $C$ in the amalgamated free pr…

Let $A$ and $B$ be finite groups, and let $C$ be a subgroup of $A\cap B$ with index at least $2$ in $A$ and at least $3$ in $B$. The Conder–Džambić–Jones conjecture. All but finite…

Interval-rank conjecture. The group $A_{11}$ is the only finite simple group whose set of ranks of string C-group representations is not an interval in the set of integers.

Let $\widetilde{A}_n$ be the double cover of the alternating group $A_n$, and let $\operatorname{Irr}(\widetilde{A}_n)$ denote its irreducible characters. A character is called spi…

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…

Alternating-group indicator conjecture. The indicator satisfies

Let $G=A_n$ and $H=(S_k\times S_{n-k})\cap G$, where $1<k<n$ and $p\leqslant k$ for every prime number $p\leqslant n$. Prime-graph equality conjecture. If $n\geqslant 12$, then … i…

Normal covering conjecture for $A_n$.

Let $G=A_n$ with $n$ odd, let $C$ be a conjugacy class of $n$-cycles, and let $H$ be the centralizer in $G$ of an element of $C$. Let $N_G(H)$ denote the normalizer of $H$ in $G$,…