888 problems

Let $G$ be a finite group, let $S$ be a generating set, let $n$ denote the group order, let $g_{\mathrm{min}}$ be the minimal number of generators of $G$, and let…

Rank-two conjecture. If $m\geqslant n\geqslant 2$ and

Let $G$ be a finite simple group of Lie type, let $p$ be a prime different from the defining characteristic of $G$, and let $m$ be a positive integer. For elements $g,g'\in G$, wri…

Let $G$ be a finite group and let $p$ be a prime dividing $|G|$. Let ${\mathcal{A}}_p(G)$ be the poset of non-trivial elementary abelian $p$-subgroups of $G$, ordered by inclusion,…

Let $G$ be a finite group, $p$ a prime, and $B$ a $p$-block of $G$. A $p$-weight of $B$ is a $p$-weight belonging to the block $B$, and the irreducible $p$-Brauer characters of $B$…

For a fixed integer $k\ge 2$, let $\Gamma$ be a finite abelian group of order $n$ with no nonzero element whose order divides $k$. Define $f_k(\Gamma)$ to be the largest integer su…

For every finite group $G$, every integer $n\ge 1$, every choice of subgroups $H_1,\ldots,H_n\le G$, and every $g_1,\ldots,g_n\in G$, if $U=\bigcup_{i=1}^n g_iH_i$ is a proper subs…

Given a finite abelian group $G$ of order $v$ and integers $k$ and $\lambda$, determine whether there exists a function $x:G\to\{-1,0,1\}$ such that $\sum_{g\in G}x(g)^2=k$ and, fo…

For every finite nonabelian group $G$, the Gao constant satisfies $E(G)=\mathsf d(G)+|G|$, where $\mathsf d(G)$ is the largest integer $\ell$ for which there exists a product-one-f…

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

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

Herzog's conjecture. If $I_2(G)=I_2(H)$, then $|G|=|H|$.

Let $k$ be a fixed positive integer and let $G$ be a finite group. Suppose that at least half of the elements of $G$ have order $k$. Deaconescu's conjecture. Then $G$ is soluble. T…

Let $G_1$ and $G_2$ be groups, and let a circular Costas map be a map $\varphi:G_1\to G_2$ that is injective and whose nonzero difference maps are injective; call it standard when…

Let $J$ be a finite-dimensional nilpotent algebra over a finite field $\operatorname{F}$, let $G=1+J$ be the associated algebra group, and let…

Alperin's Conjecture C. If $P$ is TI, then

For each of the 23 umbral cases, let $G^X$ be the corresponding finite group, let $m$ be its associated positive integer, and let $I^X\subset\{1,2,\dots,m-1\}$ be the specified sub…

Let $Q=(Q,\star)$ be a quandle, meaning that $a\star a=a$ for every $a\in Q$ and every left translation $L_a\colon Q\to Q$, $x\mapsto a\star x$, is an automorphism. The left multip…

Let $G$ be a finite group, let $p$ be a prime, let $P\in\operatorname{Syl}_p(G)$, and let $x\in P$. Let $R_p(x)$ be a set of representatives for the conjugacy classes in the reduce…

Polylogarithmic-party hardness conjecture. There is no protocol for the $k$-party $t$-tuple interleaved group product over $G$ with $k=\log^c t$ parties and communication…

Equality-case conjecture. If $S_1$ and $S_2$ are cross-1-intersecting with respect to $\beta$ and

Let $G$ be a nonabelian finite simple group, and let $\operatorname{diam}(G)$ denote the maximum of $\operatorname{diam}(G,X)$ over all generating sets $X$ of $G$. Babai’s conjectu…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, let $G$ be a finite group, and let $\mathcal{O}_G$ be its $p$-orbit c…

For a finite group $A$ and a subgroup $U\leqslant A$, let $\mathcal{P}(U)$ be the set of elements of $U$ of prime-power order, and define … the union of the $A$-conjugacy classes o…