1,836 problems

Let $\ell$ be a prime. A McKay bijection with respect to $\ell$ is a bijection between the irreducible characters of a finite group and those of an appropriate normaliser of a Sylo…

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…

Thompson's conjecture. Every finite nonabelian simple group $S$ is uniquely determined by the set $\operatorname{cs}(S)$ in the class of all finite groups with trivial center; equi…

Zassenhaus conjectures. If $H\in\mathcal{G}_i$, then there exists a subgroup $K_H\leqslant G$ such that $H$ and $K_H$ are conjugate in the unit group of ${\mathbb Q}G$.

Let $G$ be a finite group and let $B$ be a $p$-block of $G$ with defect group $D$. Brauer's $k(B)$-conjecture. The number of irreducible characters in $B$ does not exceed the order…

Let $U_n(q)$ be the group of upper-triangular $n\times n$ matrices over $\mathbb{F}_q$ with units on the diagonal; equivalently, it is a maximal unipotent subgroup of…

Donovan's conjecture. For a fixed defect group, there are only finitely many block algebras of group-block pairs up to Morita equivalence.

Let $w=[x,y]=x^{-1}y^{-1}xy$ be the commutator word, and for a group $G$ let $w(G)$ be its verbal set. The word $w$ has width $N$ in $G$ if the verbal subgroup generated by $w(G)$…

Let $G$ be a finite group. Write … and … Here $\operatorname{Irr}(G)$ is the set of complex irreducible characters of $G$, and $\pi(n)$ is the set of prime divisors of the positive…

Let $f(n)$ denote the number of isomorphism classes of finite groups of order $n$. Surjectivity conjecture. For every positive integer $m$, there exists an integer $n$ such that ……

P. Hall's conciseness conjecture. Every word is concise in the class of all groups.

Let $G$ be a finite primitive permutation group of degree $n$ with point stabiliser $H$, and let $b(G,H)$ be its base size. Pyber's conjecture. There exists an absolute constant…

Let $G\leqslant \operatorname{Sym}(\Omega)$ be a primitive almost simple permutation group in a non-standard action, meaning that its point stabiliser is non-standard. Let…

For a finite group $G$, let $L(G)$ be its Cayley table. The Hall–Paige condition is the condition that the sum of the elements of $G$ is the identity in the abelianisation…

Let $G$ be a finite group and let $p$ be a prime. Let $\mathcal{S}_p(G)$ be the poset of all non-trivial $p$-subgroups of $G$, and let $\operatorname{O}_p(G)$ be the largest normal…

Miller's conjecture. One has

Non-inner automorphism conjecture. Every finite non-abelian $p$-group has a non-inner automorphism of order $p$.

For a fixed $n\in\mathbb{N}$, let $f_n(p)$ denote the number of isomorphism classes of finite $p$-groups of order $p^n$. Higman's PORC conjecture. There exist a fixed integer $N$ a…

Mathieu moonshine conjecture. Each $\mathcal{D}_i$ is, in some natural but unexplained way, a representation of the finite simple group $M_{24}$. The conjecture concerns the appear…

Let $G$ be an abelian group, and let $A=\{a_1,\ldots,a_k\}$ and $B=\{b_1,\ldots,b_k\}$ be subsets of $G$, where $G$ has odd order. Snevily's conjecture. There is a permutation…

Arad–Herzog conjecture. The product of two nontrivial conjugacy classes of $G$ is never a conjugacy class.

Let $n$ be a positive integer, let $\lambda,\mu\vdash n$, and let $q$ be a prime power. Write $\operatorname{GL}_n(\mathbb F_q)$ for the general linear group and…

Babai–Godsil conjecture. As the order of $G$ grows, almost all finite Cayley digraphs of $G$ are DRRs.

Abhyankar's conjecture. The finite quotient groups of the étale fundamental group of $\mathbb{G}_m$ are precisely the finite groups $G$ such that

Let $H$ be a finite group, and let $\mathcal{M}(H)$ denote the set of maximal subgroups of $H$. Wall's conjecture. The bound … holds for every finite group $H$. The conjecture woul…