Mathematics Problem Archive

(Well-known problem). A finite group G is called an IYB-group if it is isomorphic to the permutation group of a finite involutive non-degenerate set-t...

(Well-known question). A discrete group G is said to have the Haagerup property (also known as Gromov's a-T-menability property) if there exists a met...

Conjecture: If N is a finite soluble group, then any regular subgroup in the holomorph Hol(N) of N is also soluble....

(Well-known problem). A group $G$ is a unique product group if, for any nonempty finite subsets $A,B$ of $G$, there exists an element of $G$ which can...

(Well-known problem). Does there exist a finitely presented (infinite) simple group requiring more than two generators?...

(Well-known problem). Does there exist a finitely presented (infinite) simple group of finite cohomological dimension greater than 2?...

(Well-known problem). Does there exist a finitely presented group $G$ such that $G\cong G\times H$ for some non-trivial group $H$?...

Let $\ell(X)$ denote the composition length of a finite group $X$. Let $A$ be a finite nilpotent group acting by automorphisms on a finite soluble gro...

A finite group $G$ is said to be semi-abelian if it has a sequence of subgroups $1=G_0\leqslant G_1\leqslant\cdots\leqslant G_n=G$ such that for every...

Let $\Gamma$ be a finite simple group and let $N_n(\Gamma)$ denote the set of normal subgroups of the free group $F_n$ of rank $n$ whose quotient is i...

Conjecture: For $n\geqslant 3$, there are no finite simple characteristic quotients of the free group $F_n$....

Conjecture: Metabelian groups are permutation-stable....

A group $G$ is said to be sofic if for every finite set $F\subseteq G$ containing $1$ and every $\varepsilon>0$ there exist $n\in\mathbb N$ and a map ...

Conjecture: The sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$ is at least the $p'$-part of $|G|$....

Conjecture: The number of irreducible $p$-Brauer characters of a finite group $G$ is bounded above by the maximum of the number of conjugacy classes $...

Conjecture: If $G$ is a transitive permutation group on a finite set $\Omega$, then for any distinct $\alpha,\beta\in\Omega$ there is an element $g\in...

For a group word $w(x_1,\ldots,x_n)$ on $n$ letters, define $e_0(x_1,\ldots,x_n)=x_1$ and $e_{k+1}(x_1,\ldots,x_n)=w(e_k(x_1,\ldots,x_n),\ldots,x_n)$ ...

Conjecture: The derived length of a finite solvable group $G$ does not exceed $|\operatorname{Cod}(G)|-1$....

Let $S$ be a nonabelian finite simple group, and $x$ a nonidentity automorphism of $S$. Let $\alpha(x)$ be the smallest number of conjugates of $x$ in...

Conjecture: Let $G$ be a finite additive abelian group with $|G|$ odd. Then any subset $A$ of $G$ with $|A|=n>2$ can be written as $\{a_1,\ldots,a_n\}...

(Well-known problem). Is Thompson's group F automatic?...

Conjecture: Every subgroup of Thompson's group F is either elementary amenable or else contains a subgroup isomorphic to F....

(Well-known problem). A classifying space for a group $G$ is a connected CW-complex with fundamental group $G$ and all higher homotopy groups trivial....

Conjecture If a group has $r$ generators and exponent $n$, is it necessarily finite?...

Conjecture Every finite group is the Galois group of some finite algebraic extension of $\mathbb Q$....

Conjecture There exists a finite lattice that is not an interval in the subgroup lattice of a finite group....

(Gromov) Does every one-ended word-hyperbolic group contain a closed hyperbolic surface subgroup?...

(Gromov) For a given n is there an example of a hyperbolic group of dimension n in which every infinite index subgroup is free? Or in which there are ...

(Swarup) Suppose $H$ is a finitely presented subgroup of a word-hyperbolic group $G$ and has finite index in its normalizer. Assume that there is $n>0...

(Whyte) Let Γ be a 1-ended hyperbolic group which is not virtually a surface group. Can every infinite index subgroup be free?...

Is every word-hyperbolic group residually finite?...

(Dani Wise) Let $G^n$ denote the Cartesian product of $n$ copies of $G$, and let $\operatorname{rank}(G^n)$ be its smallest number of generators. If $...

(Dani Wise) Find 'nice' groups, for example CAT(0) or automatic groups, for which $\operatorname{rank}(G^n)$ does not tend to infinity as $n\to\infty$...

(Ian Leary) Is there a version of the Kan-Thurston theorem using only CAT(-1) groups, or word hyperbolic groups? (The statement should be: for any fin...

(Gromov) If $G$ admits a finite-dimensional $K(G,1)$, does $G$ act properly discontinuously by isometries on a complete CAT(0) space?...

Suppose a group G acts properly discontinuously and cocompactly by isometries on two CAT(0) spaces X and Y . Croke-Kleiner have examples where the bou...

Do CAT(0) (or (bi)automatic) groups satisfy the Tits alternative?...

(Eric Swenson) Let $X$ be a proper CAT(0) metric space and $G$ a finitely generated group acting properly discontinuously by isometries on $X$. (1) Ca...

(Ruth Charney) Classify Coxeter groups up to isomorphism....

(Ruth Charney) Classify Artin groups up to isomorphism....

(Ruth Charney) Are all [finite type] Artin groups CAT(0)?...

(Ruth Charney) Are all Artin groups automatic?...

(J. Cornick) If G is f.g. and the homological dimension hd G = 1, is G free?...

Characterize 1-relator groups which are limit groups....

(Swarup) Suppose G is a 1-ended finitely presented group that acts on a compact connected metric space X as a convergence group. What can be said abou...

(Jim Anderson) If G is a group of isometries of Hn, denote by Ax(G) the set of axes of the elements of G. If G1 and G2 are finitely generated and disc...

(Misha Kapovich) Is there a finitely-generated discrete subgroup of SO(n, 1) whose action on the limit set is not ergodic? Is not recurrent?...

(Ian Leary) Is there a group of finite vcd that does not act with finite stabilizers on an acyclic complex of dimension equal to its vcd?...

(Peter Kropholler) If G is FP over the rationals, is there a bound on the orders of finite subgroups of G?...

(Sageev) Is there a f.p. 1-ended group G with G ∼= G∗Z?...