Kourovka Notebook 21

Is it true that for every positive rational number $r$ there exists a finite group $G$ such that $|\operatorname{Aut}(G)|/|G|=r$?

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 $a$ with $C_B(a)=1$. Is it true th…

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

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?

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 right-relatively convex subgroups…

(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 group is of type $F_n$ if i…

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

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

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

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 primes. Does every finite group…

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?

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 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 finite-index subgroups of G is bou…

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 surface 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 of exponent $p^2$ form a subg…

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?

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 multiplicities. Suppose that $H$ is…

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 $x^n=1$ in $G$. a) Is it true that a…

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

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-trivial finite centre.

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

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 $\{a_1,\ldots,a_n\}$ in such a way that all the s…

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

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 necessarily of the same index) such tha…

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