Unsolved Problems

Let $C_1,\ldots,C_n$ be (left or right) cosets of a finite group $G$ such that $U:=C_1\cup\cdots\cup C_n$ is not $G$. Is it always true that $|G\setmi...

A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Is every branch group boundedl...

(a) Does there exist a finitely generated simple group that is of exponential growth but not of uniformly exponential growth? (b) Does there exist a ...

Is there any group which is not isomorphic to the quotient of a residually finite group by an amenable normal subgroup?...

Does there exist a group $G$ that contains a family $(G_n)_{n\in\mathbb N}$ of finite-index subgroups such that for every $n$ there is a homomorphism ...

A pro-p group is (relatively) strictly finitely presented if it is the pro-p completion of a group that is finitely presented (respectively, finitely ...

Let $p$ be a prime number. A group $\Gamma$ is called $p$-Jordan if there exist constants $J$ and $e$ such that any finite subgroup $G\subset\Gamma$ c...

Let w be a group word, and G a profinite group. Is it true that the cardinality of the set of w-values in G is either finite or at least continuum?...

Is it true that the extension of the A. Agrachev--R. Gamkrelidze construction of groups from pre-Lie rings suggested in Definition 66 produces groups ...

A group G is said to be virtually special if G has a finite-index subgroup isomorphic to the fundamental group of a special complex. A group G is call...

Let $F_m$ be a free group of rank $m$ and let $\varphi\in\operatorname{Aut}(F_m)$ be a polynomially growing automorphism of maximal degree $m-1$, whic...

Do there exist finitely presented subgroups of right-angled Artin groups whose Dehn functions are super-exponential, or sub-exponential but not polyno...

Let G be a right-angled Artin group. Is the stable commutator length scl(g) a rational number for every g $\in$ [G, G]?...

Two groups $G_1$ and $G_2$ are said to be commensurable if there exist finite-index subgroups $H_1\leqslant G_1$ and $H_2\leqslant G_2$ (not necessari...

If two Artin groups of spherical type are quasi-isometric, must they be commensurable? (This is not true for right-angled Artin groups.)...

Construct a homomorphism of a subgroup of a Golod group onto an infinite AT-group....

Based on the development of E. S. Golod's construction, for each prime number p, construct a finitely generated residually finite p-group with a non-t...

Does a group need to have a subnormal abelian series if every countable subgroup of it has such a series?...

For a finite group $G$, let the type of $G$ be the function on positive integers whose value at $n$ is the number of solutions of the equation $x^n=1$...

For a finite group $G$, let $\chi_1(G)$ denote the totality of the degrees of all irreducible complex characters of $G$ with allowance for their multi...

Let G be a profinite group with fewer than $2^{\aleph_0}$ conjugacy classes of elements of infinite order. Must G be a torsion group?...

If the $p$-th powers in a finite $p$-group form a subgroup, must that subgroup be powerful? That is, for $p\ne 2$, if the $p$-th powers in a $p$-group...

Let G be an infinite finitely presented group such that every subgroup of infinite index is free. Must G be isomorphic to either a free group or a sur...

Let G be a hyperbolic group which is virtually compact special in the sense of Haglund--Wise. Suppose that the set of second Betti numbers of the fini...

Let G be a torsion-free group of type $F_\infty$ of infinite cohomological dimension. Must G contain a copy of Thompson's group F?...

Let $G=G_1\amalg_H G_2$ be a free pro-$p$ product of coherent pro-$p$ groups with polycyclic amalgamation. Is $G$ coherent?...

A group $G$ is said to be invariably generated by $a$ and $b$ if $G$ is generated by the conjugates $a^g,b^h$ for every $g,h$. Let $p\ne q$ be fixed p...

Is Thompson's group F quasi-isometric (a) to F $\times$ Z? (b) to F $\times$ F?...

A subgroup H of a right-orderable group G is said to be right-relatively convex if it is convex under some right ordering on G. Is the lattice of righ...

Is it true that the lattice of right-relatively convex subgroups of a right-orderable group is distributive if and only if it is a chain?...

Are there order automorphisms of Dlab groups that are not inner automorphisms?...

Let $G$ be an extension of a normal elementary abelian subgroup $A$ by an elementary abelian group $B\cong G/A$ such that $A$ contains an element $a$ ...

Problem Given a Matrix $A$, the $k$-th elementary symmetric function of $A$, namely $S_k(A)$, is defined as the sum of all $k$-by- $k$ principal mino...

Problem Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?...

A latin square is even if the product of the signs of all of the row and column permutations is 1 and is odd otherwise. Conjecture For every positive...

Given integers $k,n\ge2$, let $d(k,n)$ be the smallest integer $d\ge2$ such that the symmetric group $\frak S$ on the set of all words of length $n$ o...

Let $E$ be a non-empty finite set. Given a partition $\bf h$ of $E$, the stabilizer of $\bf h$, denoted $S(\bf h)$, is the group formed by all permuta...

Let $S_n$ denote the set of all permutations of $[n]=\set{1,2,\ldots,n}$. Let $i(\pi)$ and $d(\pi)$ denote respectively the number of increasing and t...

Conjecture If ${\mathbb F}$ is a finite field with at least 4 elements and $A$ is an invertible $n \times n$ matrix with entries in ${\mathbb F}$, the...

Conjecture For every prime $p$, there is a constant $c(p)$ (possibly $c(p)=p$ ) so that the union (as multisets) of any $c(p)$ bases of the vector spa...

Let $M$ be a matroid of rank $r$, and for $0 \le i \le r$ let $w_i$ be the number of closed sets of rank $i$. Conjecture $w_0,w_1,\ldots,w_r$ is unim...

Let $G$ be an (additive) abelian group, and for every $S \subseteq G$ let ${\mathit stab}(S) = \{ g \in G: g + S = S \}$. Conjecture Let $M$ be a mat...

Conjecture If $M_1,\ldots,M_k$ are matroids on $E$ and $\sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1)$ for every partition $\{X_1,\ldots,X_k\}$ of $E$, th...

Conjecture Let $r \ge 2$ be an integer and let $H$ be a minor minimal clutter with $\frac{1}{r}\tau_r(H) < \tau(H)$. Then either $H$ has a $J_k$ minor...

Conjecture If $A$ is 2-large, then $A$ is large....

Problem Does there exist a dense set $S \subseteq {\mathbb R}^2$ so that all pairwise distances between points in $S$ are rational?...

Question What is the minimum cardinality of a decomposition of the $n$-cube into $n$-simplices?...

Conjecture If a finite set of unit balls in $\mathbb{R}^n$ is rearranged so that the distance between each pair of centers does not decrease, then the...

Conjecture Every set of $2^{n-2} + 1$ points in the plane in general position contains a subset of $n$ points which form a convex $n$-gon....

If $X \subseteq {\mathbb R}^2$ is a finite set of points which is 2-colored, an empty triangle is a set $T \subseteq X$ with $|T|=3$ so that the conve...