Do there exist non-trivial finitely-generated divisible groups? Equivalently, do there exist non-trivial finitely-generated divisible simple groups?

Let $G$ be a group, $F$ a free group with free generators $x_1, \dots, x_n$, and $R$ the free product of $G$ and $F$. An equation (in the unknowns $x_1, \dots, x_n$) over $G$ is an expression of the form $v(x_1, \dots, x_n) = 1$, where on the left is an element of $R$ not conjugate in $R$ to any element of $G$. We call $G$ algebraically closed if every equation over $G$ has a solution in $G$.

Do there exist algebraically closed groups?

(Well-known problem). Can the group ring $\mathbb{Z}[G]$ of a torsion-free group $G$ contain zero divisors?

Does there exist a ring without zero divisors which is not embeddable in a skew field, while the multiplicative semigroup of its non-zero elements is embeddable in a group?

(Well-known problem). Does there exist a group whose group ring does not contain zero divisors and is not embeddable into a skew field?

(A. I. Mal’cev). Is the group ring of a right-ordered group embeddable into a skew field?

Can the factor-group of a locally normal group by the second term of its upper central series be embedded (isomorphically) in a direct product of finite groups?

An automorphism $\varphi$ of a group $G$ is called splitting if $g g^\varphi \dots g^{\varphi^{n-1}} = 1$ for any element $g \in G$, where $n$ is the order of $\varphi$. Is a soluble group admitting a regular splitting automorphism of prime order necessarily nilpotent?

The conjugacy problem for the braid groups $B_n$, $n > 4$.

(W. Magnus). The problem of the isomorphism to the trivial group for all groups with $n$ generators and $n$ defining relations, where $n > 2$.

(J. Stallings). If a finitely presented group is trivial, is it always possible to replace one of the defining words by a primitive element without changing the group?

(B. H. Neumann). Does there exist an infinite simple finitely presented group?

Write an explicit set of generators and defining relations for a universal finitely presented group.

a) Does there exist an algorithm for determining the solubility of equations in a free group?
b) Describe the structure of all solutions of an equation when it has at least one solution.

(A. I. Mal’cev). Which subgroups (subsets) are first order definable in a free group? Which subgroups are relatively elementarily definable in a free group? In particular, is the derived subgroup first order definable (relatively elementarily definable) in a free group?

For which groups (classes of groups) is the lattice of normal subgroups first order definable in the lattice of all subgroups?

Are there only finitely many finite simple groups of a given exponent $n$?

Does there exist an infinite simple locally finite group of finite rank?

Does every infinite group possess an infinite abelian subgroup?

a) Is the universal theory of the class of finite groups decidable?
b) Is the universal theory of the class of finite nilpotent groups decidable?

Does elementary equivalence of two finitely generated nilpotent groups imply that they are isomorphic?

Describe the universal theory of free groups.

Describe the universal theory of a free nilpotent group.

(A. Tarski). Is the elementary theory of a free group decidable?

Is the universal theory of the class of soluble groups decidable?

Is a residually finite group with the maximum condition for subgroups almost polycyclic?

Is the Frattini subgroup of a finitely generated matrix group over a field nilpotent?

(A. I. Mal’cev). Describe the automorphism group of a free solvable group.

Has every orderable polycyclic group a faithful representation by matrices over the integers?

A group is called pro-orderable if every partial ordering of the group extends to a linear ordering.
$\qquad$ a) Is the wreath product of two arbitrary pro-orderable groups again pro-orderable?
$\qquad$ b) (A. I. Mal’cev). Is every subgroup of a pro-orderable group again pro-orderable?
$\qquad$ c) (A. I. Mal’cev, L. Fuchs). Do there exist simple pro-orderable groups?

If a group $G$ is factorizable by $p$-subgroups, that is, $G = AB$, where $A$ and $B$ are $p$-subgroups, does it follow that $G$ is itself a $p$-group?

Is it true that every subgroup of a locally nilpotent group is quasi-invariant?

An $N^0$-group is a group in which every cyclic subgroup is a term of some normal system of the group.

Is every $N^0$-group an $\tilde{N}$-group?

Is a group binary nilpotent if it is the product of two normal binary nilpotent subgroups?

Is a group a nilgroup if it is the product of two normal nilsubgroups? By definition, a nilgroup is a group consisting of nilelements, in other words, of (not necessarily boundedly) Engel elements.

A subgroup of a linearly orderable group is called relatively convex if it is convex with respect to some linear ordering of the group. Under what conditions is a subgroup of an orderable group relatively convex?

Is the centre of a relatively convex subgroup relatively convex?

Is the centralizer of a relatively convex subgroup relatively convex?

Is a maximal abelian normal subgroup relatively convex?

Is the largest locally nilpotent normal subgroup relatively convex?

What conditions ensure the normalizer of a relatively convex subgroup to be relatively convex?

A subgroup $H$ of a group $G$ is said to be strictly isolated if, whenever $x g_1^{-1} x g_1 \dots g_n^{-1} x g_n$ belongs to $H$, so do $x$ and each $g_i^{-1} x g_i$. A group in which the identity subgroup is strictly isolated is called an $S$-group. Do there exist $S$-groups that are not orderable groups?

Is a free product of two orderable groups with an amalgamated subgroup that is relatively convex in each of the factors again an orderable group?

Is it possible to order an abelian strictly isolated normal subgroup of an $S$-group (see 1.47 for the definitions) in such a way that the order is preserved under the action of the inner automorphisms of the whole group?

Do the order-preserving automorphisms of a linearly ordered group form an orderable group?

What conditions ensure a matrix group over a field (of complex numbers) to be orderable?

Describe the groups that can be ordered linearly in a unique way (reversed orderings are not regarded as different).

Describe all possible linear orderings of a free nilpotent group with finitely many generators.

Describe all linear orderings of a free metabelian group with a finite number of generators.

Give an elementary classification of linearly ordered free groups with a fixed number of generators.

Is a torsion-free group pro-orderable if the factor-group by its centre is pro-orderable (see 1.35 for definition)?

Can an orderable metabelian group be embedded in a radicable orderable group?

Can any orderable group be embedded into an orderable group
$\qquad$ a) with a radicable maximal locally nilpotent normal subgroup?
$\qquad$ b) with a radicable maximal abelian normal subgroup?

A group $G$ is called dense if it has no proper isolated subgroups other than its trivial subgroup.
$\qquad$ a) Do there exist dense torsion-free groups that are not locally cyclic?
$\qquad$ b) Suppose that any two non-trivial elements $x$ and $y$ of a torsion-free group $G$ satisfy the relation $x^k = y^l$, where $k$ and $l$ are non-zero integers depending on $x$ and $y$. Does it follow that $G$ is abelian?

A torsion-free group is said to be separable if it can be represented as the set-theoretic union of two of its proper subsemigroups. Is every $R$-group separable?

Is the class of groups of abelian extensions of abelian groups closed under taking direct sums $(A, B) \mapsto A \oplus B$?

Suppose that $T$ is a periodic abelian group, and $\mathfrak{m}$ an uncountable cardinal number. Does there always exist an abelian torsion-free group $U(T, \mathfrak{m})$ of cardinality $\mathfrak{m}$ with the following property: for any abelian torsion-free group $A$ of cardinality $\leqslant \mathfrak{m}$, the equality $\text{Ext}(A, T) = 0$ holds if and only if $A$ is embeddable in $U(T, \mathfrak{m})$?

Suppose that $G$ is a finitely presented group, $F$ a free group whose rank is equal to the minimal number of generators of $G$, with a fixed homomorphism of $F$ onto $G$ with kernel $N$. Find a complete system of invariants of the factor-group of $N$ by the commutator subgroup $[F, N]$.

(A. Tarski). Let $\mathfrak{K}$ be a class of groups and $Q\mathfrak{K}$ the class of all homomorphic images of groups from $\mathfrak{K}$. If $\mathfrak{K}$ is axiomatizable, does it follow that $Q\mathfrak{K}$ is?

(B. I. Plotkin). Do there exist locally nilpotent torsion-free groups without the property $RN^*$?

Let $p$ be a prime number and let $G$ be the group of all matrices of the form $\begin{pmatrix} 1 + p\alpha & p\beta \\ p\gamma & 1 + p\delta \end{pmatrix}$, where $\alpha, \beta, \gamma, \delta$ are rational numbers with denominators coprime to $p$. Does $G$ have the property $\overline{\text{RN}}$?

Let $G$ be a connected algebraic group over an algebraically closed field. Is the number of conjugacy classes of maximal soluble subgroups of $G$ finite?

D. Hertzig has shown that a connected algebraic group over an algebraically closed field is soluble if it has a rational regular automorphism. Is this result true for an arbitrary field?

Are there only finitely many conjugacy classes of maximal periodic subgroups in a finitely generated linear group over the integers?

Describe all minimal topological groups, that is, non-discrete groups all of whose closed subgroups are discrete. The minimal locally compact groups can be described without much effort. At the same time, the problem is probably complicated in the general case.

Classify the infinite simple periodic linear groups over a field of characteristic $p > 0$.

Does there exist a simple, locally nilpotent, locally compact, topological group?

Let a group $G$ be the product of two divisible abelian $p$-groups of finite rank. Is then $G$ itself a divisible abelian $p$-group of finite rank?

Does there exist a finite simple group whose Sylow 2-subgroup is a direct product of quaternion groups?

The width of a group $G$ is, by definition, the smallest cardinal $m = m(G)$ with the property that any subgroup of $G$ generated by a finite set $S \subseteq G$ is generated by a subset of $S$ of cardinality at most $m$.
$\qquad$ a) Does a group of finite width satisfy the minimum condition for subgroups?
$\qquad$ b) Does a group with the minimum condition for subgroups have finite width?
$\qquad$ c) The same questions under the additional condition of local finiteness. In particular, is a locally finite group of finite width a Chernikov group?

Two sets of identities are said to be equivalent if they determine the same variety of groups. Construct an infinite set of identities which is not equivalent to any finite one.

Does there exist a simple group, the orders of whose elements are unbounded, in which a non-trivial identity relation holds?

Is it true that a polycyclic group $G$ is residually a finite $p$-group if and only if $G$ has a nilpotent normal torsion-free subgroup of $p$-power index?

Is it true that the identity relations of a metabelian group have a finite basis?

Is it true that the identical relations of a polycyclic group have a finite basis?

Is it true that the identical relations of a matrix group (at least over a field of characteristic 0) have a finite basis?

Is it true that if a matrix group over a field of characteristic 0 does not satisfy any non-trivial identity relation, then it contains a non-abelian free subgroup?

Is the following assertion true? Let $G$ be a free soluble group, and $a$ and $b$ elements of $G$ whose normal closures coincide. Then there is an element $x \in G$ such that $b^{\pm 1} = x^{-1}ax$.

A subgroup $H$ of a group $G$ is called 2-infinitely isolated in $G$ if, whenever the centralizer $C_G(h)$ in $G$ of some element $h \neq 1$ of $H$ contains at least one involution and intersects $H$ in an infinite subgroup, it follows that $C_G(h) \leqslant H$. Let $G$ be an infinite simple locally finite group whose Sylow 2-subgroups are Chernikov groups, and suppose $G$ has a proper 2-infinitely isolated subgroup $H$ containing some Sylow 2-subgroup of $G$. Does it follow that $G$ is isomorphic to a group of the type $\text{PSL}_2(k)$, where $k$ is a field of odd characteristic?