Unsolved Problems

Which finite almost simple groups are the automorphism groups of regular polytopes of rank 3? In other words, which finite almost simple groups are ge...

Let $V$ be a variety generated by a finite group, and let $f(n)$ be the order of the free group in $V$ on $n$ generators. Is it true that the sequence...

A Hausdorff topological group G is called minimal if it does not admit a strictly coarser Hausdorff group topology. A topological group is called Raik...

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

A group word $w$ is said to be concise in a class $\mathcal C$ of groups if for every group $G$ in $\mathcal C$ such that the set $G_w$ of word values...

A first order formula $\phi(x)$ in the group language with one free variable is said to be concise in a class $\mathcal C$ of groups if for every grou...

A sequence $\{F_n\}$ of pairwise disjoint finite subsets of a topological group is called expansive if for every open subset $U$ there is a number $m$...

For a finite group $G$ let $\operatorname{Cod}(G)$ denote the set of irreducible character codegrees of $G$ (see 20.78). Define $\sigma(G)=\max\{|\pi(...

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...

Let $S$ be a finite simple nonabelian group that is not isomorphic to any group ${}^2B_2(q)$. A nonidentity automorphism $x$ of $S$ is called a $\tau$...

A nonempty class $\mathcal X$ of finite groups is said to be complete if $\mathcal X$ is closed under taking subgroups, homomorphic images, and extens...

Let $G$ be a finite group and $p$ be a prime. Let $\Psi_{p,G}$ be the class function of $G$ which vanishes on all $p$-singular elements of $G$ and who...

A finite group G is called weakly ab-maximal if |H : [H, H]| $\leqslant$ |G : [G, G]| for all H $\leqslant$ G. Do weakly ab-maximal groups have bounde...

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.)...

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\}...

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...

(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....

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

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 $a$ ...