Classify the faithful irreducible (infinite-dimensional) representations of the nilpotent group defined by generators $a, b, c$ and relations $[a, b] = c, ac = ca, bc = cb$. (A condition for a representation to be monomial is given in (A. E. Zalesskiĭ, Math. Notes, 9 (1971), 117–123).)

(Well-known problem). Describe the automorphism group of the free associative algebra with $n$ generators, $n \geqslant 2$.

(Well-known problems).
a) Is there an algorithm that decides, for any set of group words $f_1, \dots, f_m$ (in a fixed set of variables $x_1, x_2, \dots$) and a separate word $f$, whether $f = 1$ is a consequence of $f_1 = 1, \dots, f_m = 1$?
b) Given words $f_1, \dots, f_m$, is there an algorithm that decides, for any word $f$, whether $f = 1$ is a consequence of $f_1 = 1, \dots, f_m = 1$?

Can a subgroup of a relatively free group be radicable? In particular, can a verbal subgroup of a relatively free group be radicable?

Describe the insoluble finite groups in which every soluble subgroup is either 2-closed, $2'$-closed, or isomorphic to $S_4$.

An automorphism $\sigma$ of a group $G$ is called algebraic if, for any $g \in G$, the minimal $\sigma$-invariant subgroup of $G$ containing $g$ is finitely generated. An automorphism $\sigma$ of $G$ is called an e-automorphism if, for any $\sigma$-invariant subgroups $A$ and $B$, where $A$ is a proper subgroup of $B$, $B \setminus A$ contains an element $x$ such that $[x, \sigma] \in B$. Is every e-automorphism algebraic?

Let $G$ be the free product of free groups $A$ and $B$, and $V$ the verbal subgroup of $G$ corresponding to the equation $x^4 = 1$. Is it true that, if $a \in A \setminus V$ and $b \in B \setminus V$, then $(ab)^2 \notin V$?

Does there exist an infinite periodic group with a finite maximal subgroup?

Need the number of conjugacy classes of a finite simple group be bounded by a function of its exponent?

An element $g$ of a group $G$ is said to be generalized periodic if there exist $x_1, \dots, x_n \in G$ such that $x_1^{-1} g x_1 \dots x_n^{-1} g x_n = 1$. Does there exist a finitely generated torsion-free group all of whose elements are generalized periodic?

(Well-known problem). Is a locally finite group with a full Sylow basis locally soluble?

Is the elementary theory of subgroup lattices of finite abelian groups decidable?

Let $G$ be a finite group of $n \times n$ matrices over a skew field $T$ of characteristic zero. Prove that $G$ has a soluble normal subgroup $H$ whose index in $G$ is not greater than some number $f(n)$ not depending on $G$ or $T$. This problem is related to the representation theory of finite groups (the Schur index).

A group $G$ is said to be $U$-embeddable in a class $\mathfrak{K}$ of groups if, for any finite submodel $M \subset G$, there is a group $A \in \mathfrak{K}$ such that $M$ is isomorphic to some submodel of $A$. Are the following groups $U$-embeddable in the class of finite groups:
$\qquad$ a) every group with one defining relation?
$\qquad$ b) every group defined by one relation in the variety of soluble groups of a given derived length?

The word problem for a group admitting a single defining relation in the variety of soluble groups of derived length $n$, $n \geqslant 3$.

a) Is a non-abelian group with a unique linear ordering necessarily simple?
b) Is a non-commutative orderable group simple if it has no non-trivial normal relatively convex subgroups?

(B. H. Neumann). A group $U$ is called universal for a class $\mathfrak{K}$ of groups if $U$ contains an isomorphic image of every member of $\mathfrak{K}$. Does there exist a countable group that is universal for the class of countable orderable groups?

(A. I. Mal’cev). For an arbitrary linearly ordered group $G$, does there exist a linearly ordered abelian group with the same order-type as $G$?

Does the class of orderable groups coincide with the smallest axiomatizable class containing pro-orderable groups (see 1.35)?

Let $\mathfrak{K}$ be the class of one-based models of signature $\sigma$, $\vartheta$ a property that makes sense for models in $\mathfrak{K}$, and $\mathfrak{K}_0$ the class of two-based models whose first base is a set $M$ taken from $\mathfrak{K}$ and whose second base consists of all submodels of $M$ with property $\vartheta$ and whose signature consists of the symbols in $\sigma$ together with $\in$ and $\subseteq$ in the usual set-theoretic sense. The elementary theory of $\mathfrak{K}_0$ is called the element-$\vartheta$-submodel theory of $\mathfrak{K}$. Is any of the following theories decidable:
$\qquad$ a) the element-pure-subgroup theory of abelian groups?
$\qquad$ b) the element-pure-subgroup theory of abelian torsion-free groups?
$\qquad$ c) the element-$\vartheta$-submodel theory of abelian groups, when the set of $\vartheta$-subgroups is linearly ordered by inclusion?

Let $\xi = \{G_\alpha, \pi_\beta^\alpha \mid \alpha, \beta \in I\}$ be a projective system (over a directed set $I$) of finitely generated free abelian groups. If all the projections $\pi_\beta^\alpha$ are epimorphisms and all the $G_\alpha$ are non-zero, does it follow that $\varprojlim \xi \neq 0$? Equivalently, suppose every finite set of elements of an abelian group $A$ is contained in a pure finitely-generated free subgroup of $A$. Then does it follow that $A$ has a direct summand isomorphic to the infinite cyclic group?

(F. Gross). Is it true that finite groups of exponent $p^\alpha q^\beta$ have nilpotent length $\leqslant \alpha + \beta$?

(J. G. Thompson). Is every finite simple group with a nilpotent maximal subgroup isomorphic to some $\text{PSL}_2(q)$?

If $G$ is a finite 2-group with cyclic centre and every abelian normal subgroup 2-generated, then is every abelian subgroup of $G$ 3-generated?

Under what conditions can a wreath product of matrix groups over a field be represented by matrices over a field?

A torsion-free abelian group is called factor-decomposable if, in all its factor groups, the periodic part is a direct summand. Characterize these groups.

Find necessary and sufficient conditions under which every pure subgroup of a completely decomposable torsion-free abelian group is itself completely decomposable.

Are two groups necessarily isomorphic if each of them can be defined by a single relation and is a homomorphic image of the other one?

(Well-known problem). The conjugacy problem for groups with a single defining relation.

(K. Ross). Suppose that a group $G$ admits two topologies $\sigma$ and $\tau$ yielding locally compact topological groups $G_\sigma$ and $G_\tau$. If the sets of closed subgroups in $G_\sigma$ and $G_\tau$ are the same, does it follow that $G_\sigma$ and $G_\tau$ are topologically isomorphic?

Suppose that every finitely generated subgroup of a locally compact group $G$ is pronilpotent. Then is it true that every maximal closed subgroup of $G$ contains the derived subgroup $G'$?

Describe the topological groups which have no proper closed subgroups.

Describe the finite groups with a self-centralizing subgroup of prime order.

(I. R. Shafarevich). Let $\text{SL}_2(\mathbb{Z})^\wedge$ and $\text{SL}_2(\mathbb{Z})^-$ denote the completions of $\text{SL}_2(\mathbb{Z})$ determined by all subgroups of finite index and all congruence subgroups, respectively, and let $\psi : \text{SL}_2(\mathbb{Z})^\wedge \to \text{SL}_2(\mathbb{Z})^-$ be the natural homomorphism. Is $\text{Ker}\,\psi$ a free profinite group?

Is every compact periodic group locally finite?

(Kneser–Tits conjecture). Let $G$ be a simply connected $k$-defined simple algebraic group, and $E_k(G)$ the subgroup generated by unipotent $k$-elements. If $E_k(G) \neq 1$, then $G_k = E_k(G)$. The proof is known for $k$-decomposable groups (C. Chevalley) and for local fields (V. P. Platonov).

Let $\mu$ be an infinite cardinal number. A group $G$ is said to be $\mu$-overnilpotent if every cyclic subgroup of $G$ is a member of some ascending normal series of length less than $\mu$ reaching $G$. It is not difficult to show that the class of $\mu$-overnilpotent groups is a radical class. Is it true that if $\mu_1 < \mu_2$ for two infinite cardinal numbers $\mu_1$ and $\mu_2$, then there exists a group $G$ which is $\mu_2$-overnilpotent and $\mu_1$-semisimple?

Suppose that a group is generated by its subinvariant soluble subgroups. Is it necessarily locally soluble?

Let $\mathfrak{X}$ be a hereditary radical. Is it true that, in a locally nilpotent torsion-free group $G$, the subgroup $\mathfrak{X}(G)$ is isolated?

Does there exist a group having more than one, but finitely many maximal locally soluble normal subgroups?

(Well-known problem of A. I. Maltsev). Is it true that every locally nilpotent group is a homomorphic image of some torsion-free locally nilpotent group?

It can be shown that hereditary radicals form a semigroup with respect to taking the products of classes. It is an interesting problem to find all indecomposable elements of this semigroup. In particular, we point out the problem of finding all indecomposable radicals contained in the class of locally finite $p$-groups.

Does the semigroup generated by all indecomposable radicals satisfy any identity?

Let $G$ be a group of order $p^\alpha \cdot m$, where $p$ is a prime, $p$ and $m$ are coprime, and let $k$ be an algebraically closed field of characteristic $p$. Is it true that if the indecomposable projective module corresponding to the 1-representation of $G$ has $k$-dimension $p^\alpha$, then $G$ has a Hall $p'$-subgroup? The converse is trivially true.

Is it true that every finite group with a group of automorphisms $\Phi$ which acts regularly on the set of conjugacy classes of $G$ (that is, leaves only the identity class fixed) is soluble? The answer is known to be affirmative in the case where $\Phi$ is a cyclic group generated by a regular automorphism.

Can the quasivariety generated by the free group of rank 2 be defined by a system of quasi-identities in finitely many variables?

Let $L(\mathfrak{N}_4)$ denote the lattice of subvarieties of the variety $\mathfrak{N}_4$ of nilpotent groups of class at most 4. Is $L(\mathfrak{N}_4)$ distributive?

Is every binary soluble group all of whose abelian subgroups have finite rank, locally soluble?

Is a 2-group with the minimum condition for abelian subgroups locally finite?

Determine the laws of distribution of non-soluble and simple group-theoretic numbers in the sequence of natural numbers. See 2.78.

Let $G$ be a compact 0-dimensional topological group all of whose Sylow $p$-subgroups are direct products of cyclic groups of order $p$. Then is every normal subgroup of $G$ complementable?

Let $H$ be an insoluble minimal normal subgroup of a finite group $G$, and suppose that $H$ has cyclic Sylow $p$-subgroups for every prime $p$ dividing $|G : H|$. Prove that $H$ has at least one complement in $G$.

The notion of the $p$-length of an arbitrary finite group was introduced in (L. A. Shemetkov, Math. USSR Sbornik, 1 (1968), 83–92). Investigate the relations between the $p$-length of a finite group and the invariants $c_p, d_p, e_p$ of its Sylow $p$-subgroup.

Let $\sigma$ be an automorphism of prime order $p$ of a finite group $G$, which has a Hall $\pi$-subgroup with cyclic Sylow subgroups. Suppose that $p \in \pi$. Does the centralizer $C_G(\sigma)$ have at least one Hall $\pi$-subgroup?

(Well-known problem). A finite group is said to be a $D_\pi$-group if any two of its maximal $\pi$-subgroups are conjugate. Is an extension of a $D_\pi$-group by a $D_\pi$-group always a $D_\pi$-group?

Describe the finite simple groups with a Sylow 2-subgroup of the following type: $\langle a, t \mid a^{2^n} = t^2 = 1, tat = a^{2^{n-1}-1} \rangle$, $n > 2$.