Unsolved Problems

(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...

Conjecture: Suppose that for a fixed positive integer $k$ at least half of the elements of a finite group $G$ have order $k$. Then $G$ is solvable....

(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....