Classify the finite groups having a Sylow $p$-subgroup as a maximal subgroup.

A quasigroup is a groupoid $Q(\cdot)$ in which the equations $ax = b$ and $ya = b$ have a unique solution for any $a, b \in Q$. Two quasigroups $Q(\cdot)$ and $Q(\circ)$ are isotopic if there are bijections $\alpha, \beta, \gamma$ of the set $Q$ onto itself such that $x \circ y = \gamma(\alpha x \cdot \beta y)$ for all $x, y \in Q$. It is well-known that all quasigroups that are isotopic to groups form a variety $\mathfrak{G}$. Let $\mathfrak{V}$ be a variety of quasigroups. Characterize the class of groups isotopic to quasigroups in $\mathfrak{G} \cap \mathfrak{V}$. For which identities characterizing $\mathfrak{V}$ is every group isotopic to a quasigroup in $\mathfrak{G} \cap \mathfrak{V}$? Under what conditions on $\mathfrak{V}$ does any group isotopic to a quasigroup in $\mathfrak{G} \cap \mathfrak{V}$ consist of a single element?

A finite group is called quasi-nilpotent (resp. $\Gamma$-quasi-nilpotent) if any two of its subgroups (resp. maximal subgroups) $A$ and $B$ satisfy one of the conditions
$\qquad$ 1) $A \leqslant B$,
$\qquad$ 2) $B \leqslant A$,
$\qquad$ 3) $N_A(A \cap B) \neq A \cap B \neq N_B(A \cap B)$.
Do the classes of quasi-nilpotent and $\Gamma$-quasi-nilpotent groups coincide?

(S. Chase). Suppose that an abelian group $A$ can be written as the union of pure subgroups $A_\alpha$, $\alpha \in \Omega$, where $\Omega$ is the first non-denumerable ordinal, $A_\alpha$ is a free abelian group of denumerable rank, and, if $\beta < \alpha$, then $A_\beta$ is a direct summand of $A_\alpha$. Does it follow that $A$ is a free abelian group?

According to Plotkin, a group is called an $NR$-group if the set of its nil-elements coincides with the locally nilpotent radical, or, which is equivalent, if every inner automorphism of it is locally stable. Can an $NR$-group have a nil-automorphism that is not locally stable?

In an $NR$-group (see 2.5), the set of generalized central elements coincides with the nil-kernel. Is the converse true, that is, must a group be an $NR$-group if the set of generalized central elements coincides with the nil-kernel?

Find the cardinality of the set of all polyverbal operations acting on the class of all groups.

(A. I. Mal’cev). Do there exist regular associative operations having the heredity property with respect to transition from the factors to their subgroups?

Do there exist regular associative operations on the class of groups satisfying the weakened Mal’cev condition (that is, monomorphisms of the factors of an arbitrary product can be glued together, generally speaking, into a homomorphism of the whole product), but not satisfying the analogous condition for epimorphisms of the factors?

Prove an analogue of the Remak–Shmidt theorem for decompositions of a group into nilpotent products.

(Well-known problem). Let $G$ be a torsion group in which every $\pi$-element commutes with every $\pi'$-element. Does $G$ decompose into the direct product of a maximal $\pi$-subgroup and a maximal $\pi'$-subgroup?

Does there exist a torsion-free group such that the factor group by some term of its upper central series is nontrivial periodic, with a bound on the orders of the elements?

A group $G$ is called congruacy separable if any two of its elements are conjugate in $G$ if and only if their images are conjugate in every finite homomorphic image of $G$. Is $G$ conjugacy separable in the following cases:
$\qquad$ a) $G$ is a polycyclic group,
$\qquad$ b) $G$ is a free soluble group,
$\qquad$ c) $G$ is a group of (all) integral matrices,
$\qquad$ d) $G$ is a finitely generated group of matrices,
$\qquad$ e) $G$ is a finitely generated metabelian group?

Is it true that the wreath product $A \wr B$ of two groups that are conjugacy separable is itself conjugacy separable if and only if either $A$ is abelian or $B$ is finite?

Compute the ranks of the factors of the lower central series of a free soluble group.

Are finitely generated subgroups of a free soluble group finitely separable?

Is it true that the wreath products $A \wr B$ and $A_1 \wr B_1$ are elementarily equivalent if and only if $A, B$ are elementarily equivalent to $A_1, B_1$, respectively?

Do the classes of Baer and Fitting groups coincide?

An abstract group-theoretic property $\Sigma$ is said to be radical (in our sense) if, in any group $G$, the subgroup $\Sigma(G)$ generated by all normal $\Sigma$-subgroups is a $\Sigma$-subgroup itself (called the $\Sigma$-radical of $G$). A radical property $\Sigma$ is said to be strongly radical if, for any group $G$, the factor-group $G/\Sigma(G)$ contains no non-trivial normal $\Sigma$-subgroups.
$\qquad$ a) Is the property $\overline{RN}$ radical? strongly radical?
$\qquad$ b) Is the property $N^0$ (see 1.38) radical?

A subgroup $H$ of a group $G$ is called quasisubinvariant if there is a normal system of $G$ passing through $H$. Let $\mathfrak{K}$ be a class of groups that is closed with respect to taking homomorphic images. A group $G$ is called an $R^0(\mathfrak{K})$-group if each of its non-trivial homomorphic images has a non-trivial quasisubinvariant $\mathfrak{K}$-subgroup. Do $R^0(\mathfrak{K})$ and $\overline{RN}$ coincide when $\mathfrak{K}$ is the class of all abelian groups?

Are Engel torsion-free groups orderable?

a) (L. Fuchs). Describe the groups which are linearly orderable in only finitely many ways.
b) Do there exist groups that are linearly orderable in countably many ways?

(L. Fuchs). Characterize as abstract groups the multiplicative groups of orderable skew fields.

Can every orderable group be embedded in a pro-orderable group (see 1.35)?

Does the class of finite groups in which every proper abelian subgroup is contained in a proper normal subgroup coincide with the class of finite groups in which every proper abelian subgroup is contained in a proper normal subgroup of prime index?

Does there exist a finite group in which a Sylow $p$-subgroup is covered by other Sylow $p$-subgroups?

Can every group admitting an ordering with only finitely many convex subgroups be represented by matrices over a field?

Is a direct summand of a direct sum of finitely generated modules over a Noetherian ring again a direct sum of finitely generated modules?

An inverse spectrum $\xi$ of abelian groups is said to be acyclic if $\varprojlim^{(p)} \xi = 0$ for $p > 0$. Here $\varprojlim^{(p)}$ denotes the right derived functor of the projective limit functor. Let $\xi$ be an acyclic spectrum of finitely generated groups. Is the spectrum $\bigoplus \xi_\alpha$ also acyclic, where each spectrum $\xi_\alpha$ coincides with $\xi$?

(de Groot). Is the group of all continuous integer-valued functions on a compact space free abelian?

Describe the finite simple groups whose Sylow $p$-subgroups are cyclic for all odd $p$.

(Old problem). The class of rings embeddable in associative division rings is universally axiomatizable. Is it finitely axiomatizable?

Does there exist a non-finitely-axiomatizable variety of
$\qquad$ a) (H. Neumann) groups?
$\qquad$ b) of associative rings (the Specht problem)?
$\qquad$ c) Lie rings?

The $I$-theory ($Q$-theory) of a class $\mathfrak{K}$ of universal algebras is the totality of all identities (quasi-identities) that are valid on all the algebras in $\mathfrak{K}$. Does there exist a finitely axiomatizable variety of
$\quad$ a) groups,
$\quad$ b) semigroups,
$\quad$ c) $\ $(1) rings
$\quad\quad\ $ (2) of associative rings
$\quad\quad\qquad$ (i) whose $I$-theory is non-decidable?
$\quad\quad\qquad$ (ii) whose $Q$-theory is non-decidable?
$\quad\quad\ $ (3) of Lie rings
$\quad\quad\qquad$ (i) whose $I$-theory is non-decidable?
$\quad\quad\qquad$ (ii) whose $Q$-theory is non-decidable?

Is the variety generated by
$\qquad$ a) a finite associative ring;
$\qquad$ b) a finite Lie ring;
$\qquad$ c) a finite quasigroup
finitely axiomatizable?
$\qquad$ d) What is the cardinality $n$ of the smallest semigroup generating a non-finitely axiomatizable variety?

What is the structure of the groupoid of quasivarieties
$\qquad$a) of all semigroups?
$\qquad$b) of all rings?
$\qquad$c) of all associative rings?
Compare with A. I. Mal’cev, Siberian Math. J., 8, no. 2 (1967), 254–267).

A group $G$ is called an FN-group if the groups $\gamma_i G / \gamma_{i+1} G$ are free abelian and $\bigcap_{i=1}^\infty \gamma_i G = 1$, where $\gamma_{i+1} G = [\gamma_i G, G]$. A variety of groups $\mathfrak{M}$ is called a $\Sigma$-variety (where $\Sigma$ is an abstract property) if the $\mathfrak{M}$-free groups have the property $\Sigma$.
$\qquad$ a) Which properties $\Sigma$ are preserved under multiplication and intersection of varieties? Is the $FN$ property preserved under these operations?
$\qquad$ b) Are all varieties obtained by multiplication and intersection from the nilpotent varieties $\mathfrak{N}_1, \mathfrak{N}_2, \dots$ (where $\mathfrak{N}_1$ is the variety of abelian groups) $FN$-varieties?

Let $\mathfrak{A}$ and $\mathfrak{B}$ be subvarieties of a variety of groups $\mathfrak{M}$; then $(\mathfrak{A}\mathfrak{B}) \cap \mathfrak{M}$ is called the $\mathfrak{M}$-product of $\mathfrak{A}$ by $\mathfrak{B}$, where $\mathfrak{A}\mathfrak{B}$ is the usual product. Does there exist a non-abelian variety $\mathfrak{M}$ with an infinite lattice of subvarieties and commutative $\mathfrak{M}$-multiplication?

(P. Hall). Prove or refute the following conjectures:
$\qquad$ a) If a word $v$ takes only finitely many values in a group $G$, then the verbal subgroup $vG$ is finite.
$\qquad$ b) If the marginal subgroup $v^*G$ has finite index $m$ in $G$, then the order of $vG$ is finite and divides a power of $m$.
$\qquad$ c) If $G$ satisfies the maximum condition and $vG$ is finite, then $v^*G$ has finite index in $G$.

Find conditions under which a finitely-generated matrix group is almost residually a finite $p$-group for some prime $p$.

In which abelian groups is the lattice of all fully invariant subgroups a chain?

(N. Aronszajn). Let $G$ be a connected topological group locally satisfying some identical relation $f|_U = 1$, where $U$ is a neighborhood of the identity element of $G$. Is it then true that $f|_G = 1$?

(A. Selberg). Let $G$ be a connected semisimple linear Lie group whose corresponding symmetric space has rank greater than 1, and let $\Gamma$ be an irreducible discrete subgroup of $G$ such that $G/\Gamma$ has finite volume. Does it follow that $\Gamma$ is an arithmetic subgroup?

(A. Selberg). Let $\Gamma$ be an irreducible discrete subgroup of a connected Lie group $G$ such that the factor space $G/\Gamma$ is non-compact but has finite volume in the Haar measure. Prove that $\Gamma$ contains a non-trivial unipotent element.

(A. Borel, R. Steinberg). Let $G$ be a semisimple algebraic group and $R_G$ the set of classes of conjugate unipotent elements of $G$. Is $R_G$ finite?

(F. Bruhat, N. Iwahori, M. Matsumoto). Let $G$ be a semisimple algebraic group over a locally compact, totally disconnected field. Do the maximal compact subgroups of $G$ fall into finitely many conjugacy classes? If so, estimate this number.

Can $\text{SL}(n, k)$ have maximal subgroups that are not closed in the Zariski topology?

Does $\text{SL}_n(\mathbb{Z})$, $n \geqslant 2$, have maximal subgroups of infinite index?

Classify up to isomorphism the abelian connected algebraic unipotent linear groups over a field of positive characteristic. This is not difficult in the case of a field of characteristic zero. On the other hand, C. Chevalley has solved the classification problem for such groups up to isogeny.

Let $G$ be a vector space and $\Gamma$ a group of automorphisms of $G$. $\Gamma$ is called locally finitely stable if, for any finitely generated subgroup $\Delta$ of $\Gamma$, $G$ has a finite series stable relative to $\Delta$. If the characteristic of the field is zero and $\Gamma$ is locally finitely stable, then $\Gamma$ is locally nilpotent and torsion-free. Is it true that every locally nilpotent torsion-free group can be realized in this way?

Let $\Gamma$ be any nilpotent group of class $n - 1$. Does $\Gamma$ always admit a faithful representation as a group of automorphisms of an abelian group with a series of length $n$ stable relative to $\Gamma$?

Let $\Gamma$ be a Noetherian group of automorphisms of a vector space $G$ such that every element of $\Gamma$ is unipotent. Is $\Gamma$ necessarily a stable group of automorphisms?

If $G$ is a finite-dimensional vector space over a field and $\Gamma$ is a group of automorphisms of $G$ in which every element is stable, then the whole of $\Gamma$ is stable (E. Kolchin). Is Kolchin’s theorem true for spaces over skew fields?

Let $G$ be a group of automorphisms of a vector space over a field of characteristic zero, and suppose that all elements of $G$ are unipotent, with uniformly bounded unipotency indices. Must such a group be locally finitely stable (see 2.58)?

Does the set of nil-elements of a finite-dimensional linear group coincide with its locally nilpotent radical?

Does the adjoint group of a radical ring (in the sense of Jacobson) have a central series?

Is an $R$-group determined by its subgroup lattice? Is every lattice isomorphism of $R$-groups induced by a group isomorphism?

Find conditions which ensure that the nilpotent product of pure nilpotent groups (from certain classes) is determined by the lattice of its subgroups. This is known to be true if the product is torsion-free.

What can one say about lattice isomorphisms of a pure soluble group? Is such a group strictly determined by its lattice? It is well-known that the answer is affirmative for free soluble groups.

Let a group $G$ be the product of two subgroups $A$ and $B$, each of which is nilpotent and satisfies the minimum condition. Prove or refute the following:
$\qquad$ a) $G$ is soluble;
$\qquad$ b) the divisible parts of $A$ and $B$ commute elementwise.

a) Let a group $G$ be the product of two subgroups $A$ and $B$, each of which is locally cyclic and torsion-free. Prove that either $A$ or $B$ has a non-trivial subgroup that is normal in $G$.

b) Characterize the groups that can be factorized in this way.

Does there exist a finitely generated right-orderable group which coincides with its derived subgroup and, therefore, does not have the property $RN$?

(G. Baumslag). Suppose that $F$ is a finitely generated free group, $N$ its normal subgroup and $V$ a fully invariant subgroup of $N$. Is $F/V$ necessarily Hopfian if $F/N$ is Hopfian?

(Well-known problem). Does there exist an infinite group all of whose proper subgroups have prime order?

(Well-known problem). Describe the finite groups all of whose involutions have soluble centralizers.

Let $G$ be a periodic group containing an infinite family of finite subgroups whose intersection contains non-trivial elements. Does $G$ contain a non-trivial element with infinite centralizer?

Let $\Gamma$ be the holomorph of an abelian group $A$. Find conditions for $A$ to be maximal among the locally nilpotent subgroups of $\Gamma$.

Let $A$ and $B$ be abelian groups. Find conditions under which every extension of $A$ by $B$ is nilpotent.

Any set of all subgroups of the same given order of a finite group $G$ that contains at least one non-normal subgroup is called an $IE_{\bar{n}}$-system of $G$. A positive integer $k$ is called a soluble (non-soluble; simple; composite; absolutely simple) group-theoretic number if every finite group having exactly $k$ $IE_{\bar{n}}$-systems is soluble (respectively, if there is at least one non-soluble finite group having $k$ $IE_{\bar{n}}$-systems; if there is at least one simple finite group having $k$ $IE_{\bar{n}}$-systems; if there are no simple finite groups having $k$ $IE_{\bar{n}}$-systems; if there is at least one simple finite group having $k$ $IE_{\bar{n}}$-systems and there are no non-soluble non-simple finite groups having $k$ $IE_{\bar{n}}$-systems).

Are the sets of all soluble and of all absolutely simple group-theoretic numbers finite or infinite? Do there exist composite, but not soluble group-theoretic numbers?

Do there exist divisible (simple) groups with maximal subgroups?

Does every non-trivial group satisfying the normalizer condition contain a non-trivial abelian normal subgroup?

a) Does there exist an axiomatizable class of lattices $\mathfrak{K}$ such that the lattice of all subsemigroups of a semigroup $S$ is isomorphic to some lattice in $\mathfrak{K}$ if and only if $S$ is a free group?
b) The same question for free abelian groups.

Analogous questions have affirmative answers for torsion-free groups, for non-periodic groups, for abelian torsion-free groups, for abelian non-periodic groups, for orderable groups (the corresponding classes of lattices are even finitely axiomatizable). Thus, in posed questions one may assume from the outset that the semigroup $S$ is a torsion-free group (respectively, a torsion-free abelian group).

Can the class of groups with the $n$-th Engel condition $[x, \underbrace{y, \dots, y}_n] = 1$ be defined by identical relations of the form $u = v$, where $u$ and $v$ are words without negative powers of variables?

Suppose that a periodic group $G$ is the product of two locally finite subgroups. Is then $G$ locally finite?

Suppose that a locally finite group $G$ is a product of two locally nilpotent subgroups. Is $G$ necessarily locally soluble?

Is every Hall $\pi$-subgroup of an arbitrary group a maximal $\pi$-subgroup?