884 problems

For every finite group $G$ and prime $p$, let $P\in\operatorname{Syl}_p(G)$ be nonabelian, and define $m(P)=\min\{\varphi(1):\varphi\in\operatorname{Irr}(P),\ \varphi(1)>1\}$. Then…

For every finite group $G$, every prime $p$, and every Sylow $p$-subgroup $P\in\operatorname{Syl}_p(G)$, define…

For every finite-dimensional semisimple and cosemisimple Hopf algebra $H$ over a field $k$, the exponent of $H$ divides its dimension: $\exp(H)\mid\dim_k H$.

Picky Conjecture. There exists a bijection

Uniform boundedness conjecture. There exists an absolute constant $c$ such that $\rho_0(G)<c$ for all finite primitive permutation groups $G$. This conjecture asserts that the larg…

Let $p$ be a prime, and suppose that $G=P\times A$ is a finite group, where $P\in\operatorname{Syl}_p(G)$ is non-abelian and $A$ is an abelian $p'$-group. Let $\Gamma_K$ denote the…

Let $G$ be a finite non-abelian nilpotent group, and let $H$ be a group. For a finite non-abelian group $K$, let $\Gamma_K$ be the graph whose vertices are the non-central elements…

Let $G$ and $H$ be non-abelian finite groups. For a finite non-abelian group $K$, let $\Gamma_K$ be the graph whose vertices are the non-central elements of $K$, with two vertices…

Ma–Zha–She's quotient-group conjecture. There is a generator $t\langle -1,2\rangle$ of $\mathbb{Z}_p^{\times}/\langle -1,2\rangle$, for some $t\in\mathbb{Z}_p^{\times}$, such that

Let $\pi$ be a fixed subset of the set $\mathbb{P}$ of all primes with $\pi'=\mathbb{P}\setminus\pi\ne\varnothing$. A $\pi$-group is a finite group whose order has all prime diviso…

Let $f$ be the function on finite groups studied in the paper, with $f(G)=1$ precisely for cyclic groups. Second-smallest-value conjecture. $2$ is the second smallest value of the…

Zassenhaus conjecture. If $u\in\mathcal{U}(\mathbb ZG)$ is a unit of finite order, then $u$ is conjugate within $\mathbb QG$ to $\pm g$ for some $g\in G$.

Let $J_n$ be the cactus group on $n$ strands. A finite group embeds into $J_n$ precisely when it embeds into an iterated permutational wreath product of copies of…

Mann's conjecture. The quantity $|\mu(H,G)|$ is bounded by a polynomial function of $|G:H|$, and the number of subgroups $H$ of $G$ of index $m$ satisfying $\mu(H,G)\neq 0$ grows a…

Xu's conjecture. The proportion of inverse-closed subsets $S$ of $R$ such that $\operatorname{Cay}(R,S)$ is a normal Cayley graph of $R$ approaches $1$ as $r$ approaches infinity.

Babai–Godsil–Imrich–Lovász conjecture. The proportion of inverse-closed subsets $S$ of $R$ such that $\operatorname{Cay}(R,S)$ is a GRR of $R$ approaches $1$ as $r$ approaches infi…

Let $G$ be a finite group and let $\chi$ be an orthogonally stable rational character of $G$. Its orthogonal discriminant is the square class represented by a unique square-free in…

Let $G$ be a finite group, let $m(G)$ be the largest size of a minimal generating set of $G$, and let … where $\pi(G)$ is the set of prime divisors of $|G|$ and $d_p(G)$ is the min…

Let $H$ be a finite group, let $B$ be an $ell$-block of $H$, and let $D$ be a defect group of $B$. The sectional $ell$-rank $s(D)$ is the maximum of the ranks of elementary abelian…

Generalized Erdős–Ginzburg–Ziv conjecture. For every finite group $G$,

Let $G$ be a finite group, let $\mathscr{M}_G$ denote the space of admissible functions on $G$ up to rescaling, and let $\Delta_G\subset\mathscr{M}_G$ be the locus of lattice system…

Let $S_m$ denote the symmetric group of degree $m$. A finite group is OD-characterizable if it is uniquely determined, among finite groups, by its order and degree pattern; it is…

Let $L$ be a finite nonabelian simple group, and let $G$ be a finite group. The groups $G$ and $L$ are isospectral when $\omega(G)=\omega(L)$, where $\omega(H)$ denotes the set of…

The conjecture. For every $\lambda\in\mathfrak{u}_n(q)^*$,

The product conjecture. For every positive integer $k$, there exists a positive integer $n$ such that some scaling of $X^n$ is $k$-Ramsey for $X$.