186 problems

Let $p\ne q$ be fixed primes. Does every finite group embed into a finite group invariably generated by an element of order $p$ and an element of order $q$?

Are there any finite simple groups of Lie type which are totally $3$-closed? If so, find them all.

Let $n>1$ be odd, let $R_n=\mathbb{Z}/n\mathbb{Z}$ with dihedral quandle operation $a*b=2b-a$, let $\Delta(R_n)$ be the augmentation ideal of its integral quandle ring, and let…

Let $R$ be an unramified regular local ring, let $K=\operatorname{Frac}(R)$ be its fraction field, and let $G$ be a reductive group scheme over $R$. If a $G$-torsor over $R$ become…

Let $A$ and $B$ be finite essential adjacency matrices, and let $L_k(A)$ and $L_k(B)$ denote the corresponding Leavitt path algebras over a field $k$. Hazrat's graded classificatio…

Let $H_n(q)$ be the generic Iwahori--Hecke algebra of the symmetric group $d504_n$ with standard basis $T_w$ indexed by $w\in \mathfrak{S}_n$. For $1\leq i\leq \frac{n(n-1)}{2}$, d…

Let $\mathbb{K}$ be the coefficient field. Matkowski’s conjecture asserts that every two-variable formal power series $M\in\mathbb{K}[[x,y]]$ satisfying the weak associativity equa…

For every finite Latin quandle $Q$, let $\mathbb{Z}[Q]$ be the free abelian group with basis $Q$, equipped with the bilinear multiplication extending the quandle operation $x*y$ on…

For every finite associative unital ring $R$, let $N(R):=\{f(x)\in R[x]\mid f(a)=0\text{ for every }a\in R\}$ be the null ideal of $R$. The conjecture asserts that $N(R)$ is a two-…

For every finite field $k$ of characteristic $p>0$ and every integer $n\ge 3$, the tame polynomial automorphism group $\operatorname{TA}_n(k)$ is not finitely generated; equivalent…

For every fixed integer $m\ge 1$ and every collection of pairwise distinct nonconstant polynomials $f_1,\ldots,f_m\in\mathbb{C}[x]$, there exists an integer $d_0$ such that, for ev…

Let $k$ be a field and let $A$ be a finite-dimensional $k$-algebra of finite global dimension. Choose representatives $P_1,\ldots,P_n$ of the isomorphism classes of indecomposable…

For every finite-dimensional representation-finite algebra $A$ and every integer $i\ge 0$, the number of quotient-closed full additive subcategories…

Let $q$ be a prime power, let $\mathbb{F}_q$ be the field with $q$ elements, and let $\overline{\mathbb{F}_q}$ denote an algebraic closure of $\mathbb{F}_q$. For…

In mathematics, the Suita conjecture is a conjecture related to the theory of the Riemann surface, the boundary behavior of conformal maps, the theory of Bergman kernel, and the th…

Fix an integer $n \ge 1$ and let $A = \mathbb{R}[x_1,\dots,x_n]$. A subset $S \subseteq \mathbb{R}^n$ is semialgebraic if it is a finite union of sets of the form … with…

Let $n \ge 1$ and let $A \in \mathbb{C}^{n \times n}$. Write … for the field of values (numerical range) of $A$, a compact convex subset of $\mathbb{C}$, and let $\|\cdot\|$ denote…

Must every polynomial endomorphism of a rank-$n$ symplectic Poisson algebra that preserves the canonical bracket be an automorphism?

Conjecture 1.3, Section 1.4: For any root system $\Phi$, let $\xi_\Phi(s):=K_\Phi^{-s}\zeta_\Phi(s)$ be the normalized Witten zeta function. The residue of $\xi_\Phi(s)$ at its abs…

Is the depth of every finite-group cohomology ring realized by the dimension of one of its associated primes?

If the power graph of a finite group contains no induced path on four vertices, must it also contain no induced cycle of length at least four?

If $\operatorname{CT}_{(k)}$ is generated by all horizontal class transpositions with modulus at most $k$, is $\operatorname{CT}_{(k)}\cong S_{\operatorname{lcm}(2,\ldots,k)}$ for…

Given positive integers $n$ and $1\leq k\leq n!$, can $n$ distinct group elements have exactly $k$ distinct values among all their permuted products?

If $\operatorname{Sol}_G(x)$ is a proper subgroup of a nonsolvable finite group $G$, must its intersection with its normalizer be metabelian?

For a pure O-sequence $h=(h_0,\ldots,h_e)$ of codimension three and type two, is $h_i^2\geq h_{i-1}h_{i+1}$ for every interior index $i$?