1,053 problems

In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently by Namazi & Souto (2012) and…

In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The…

In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems ar…

It is an unresolved problem in the mathematical field of group theory to determine whether or not two Coxeter groups (specified by their Coxeter diagrams) are isomorphic as abstrac…

In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers $\mathbb {Q}$.…

In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974.

The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, m…

In mathematics, the Andrews–Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen t…

Strong Atiyah conjecture. If $\operatorname{lcm}(G)$ is finite, then

Let $G$ be a torsion-free group, and let $\operatorname{Wh}(G)$ denote its Whitehead group. Whitehead conjecture. The Whitehead group of any torsion-free group should vanish: … Thi…

A finitely generated group is a group generated by finitely many elements, and it has solvable word problem when there is an algorithm deciding whether any word in its generators r…

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

Atiyah's conjecture. The rank gradient $\beta_0^G(M)$ is an integer. This is a pro-$p$ analogue of the Atiyah conjecture concerning integrality of $L^2$-Betti numbers. The supplied…

Let $H$ be a coherent hyperbolic group, and let $G$ be an extension of $H$ by $\mathbb{Z}$. Wise's coherence conjecture. The group $G$ is coherent. The source attributes this conje…

Let $G$ be a nonabelian group, and call an ordering $g_1,g_2,\ldots,g_n$ of elements of $G$ valid if the partial products … are all distinct. Keedwell's conjecture. Every sufficien…

A group is co-context-free if it admits a finitely generated presentation whose set of words representing the identity is a context-free language. Lehnert's conjecture. Every co-co…

Let $D$ be a division ring, and write $D^*$ for its multiplicative group of nonzero elements. Lichtman's conjecture. If $D$ is noncommutative, then $D^*$ contains a non-cyclic free…

Let $G$ be a group with exponential growth, and let $\varphi:G\longrightarrow G$ be an injective group endomorphism. The Reidemeister number $R(\varphi)$ is the number of $\varphi$…

Let $G$ be a group. Assume $H_1,\ldots,H_k$ are proper subgroups of $G$, with $k \geq 2$, and let $x_1,\ldots,x_k$ be elements of $G$ such that … and…

Residual torsion-free nilpotence conjecture. The commutator subgroup of $\pi_1(M_J)$ is residually torsion-free nilpotent if and only if $\Delta_J$ is nontrivial.

Let $G$ be a $\mathrm{PD}^n$-group that is RFRS. Here $\beta_i^{(2)}(G;\mathbb K)$ denotes the $i$th $L^2$-Betti number with coefficients in a field $\mathbb K$. Poincaré-duality-g…

Kropholler's conjecture. If there is a proper $H$-almost invariant set $A$ such that $A=AH$, then $G$ has a non-trivial action on a tree in which $H$ fixes a vertex $v$ and every e…

Herstein's conjecture. In any characteristic, every subgroup of $D^\times$ of odd order is cyclic.

Kervaire–Laudenbach conjecture. If $G$ is nontrivial, then

Hamiltonian conjecture for Cayley graphs. Every connected Cayley graph on a finite group is Hamiltonian.