a) Is every locally finite minimal non-$FC$-group non-simple?
b) Is every locally finite minimal non-$FC$-group distinct from its derived subgroup? The question has an affirmative answer for minimal non-$BFC$-groups.

Is it true that the growth function $f$ of any infinite finitely generated group satisfies the inequality $f(n) \leqslant (f(n-1) + f(n+1))/2$ for all sufficiently large $n$ (for a fixed finite system of generators)?

(Well-known problem). Can every finite lattice $L$ be embedded in the lattice of subgroups of a finite group?

Let $g$ and $h$ be positive elements of a linearly ordered group $G$. Can one always embed $G$ in a linearly ordered group $\overline{G}$ in such a way that $g$ and $h$ are conjugate in $\overline{G}$?

If $G$ is a finitely generated abelian-by-polycyclic-by-finite group, does there exist a finitely generated metabelian group $M$ such that $G$ is isomorphic to a subgroup of the automorphism group of $M$? If so, many of the tricky properties of $G$ like its residual finiteness would become transparent.

If $R$ is a finitely generated integral domain of characteristic $p > 0$, does the profinite (ideal) topology of $R$ induce the profinite topology on the group of units of $R$? It does for $p = 0$.

An algebraic variety $X$ over a field $k$ is called rational if the field of functions $k(X)$ is purely transcendental over $k$, and it is called stably rational if $k(X)$ becomes purely transcendental after adjoining finitely many independent variables. Let $T$ be a stably rational torus over a field $k$. Is $T$ rational? Other formulations of this question and some related results see in (V. E. Voskresenskiĭ, Russ. Math. Surveys, 28, no. 4 (1973), 79–105).

Let $G$ be a torsion-free soluble group of finite cohomological dimension $\text{cd}\,G$. If $hG$ denotes the Hirsch number of $G$, then it is known that $hG \leqslant \text{cd}\,G \leqslant hG+ 1$. Find a purely group-theoretic criterion for $hG = \text{cd}\,G$.

Let $1 \to R_i \xrightarrow{\pi_i} F \to G \to 1$, $i = 1, 2$, be two exact sequences of groups with $G$ finite and $F$ free of finite rank $d(F)$. If we assume that $d(F) = d(G) + 1$ (where $d(G)$ is the minimum number of generators of $G$), are the corresponding abelianized extensions isomorphic?

Let $E = H \ast K$ and denote the augmentation ideals of the groups $E, H, K$ by $\mathfrak{e}, \mathfrak{h}, \mathfrak{k}$, respectively. If $I$ is a right ideal in $\mathbb{Z}E$, let $d_E(I)$ denote the minimum number of generators of $I$ as a right ideal. Assuming $H$ and $K$ finitely generated, is it true that $d_E(\mathfrak{e}) = d_E(\mathfrak{h}E) + d_E(\mathfrak{k}E)$? Here $\mathfrak{h}E$ is, of course, the right ideal generated by $\mathfrak{h}$; similarly for $\mathfrak{k}E$.

Let $G$ be a finite group and suppose that there exists a non-empty proper subset $\pi$ of the set of all primes dividing $|G|$ such that the centralizer of every non-trivial $\pi$-element is a $\pi$-subgroup. Does it follow that $G$ contains a subgroup $U$ such that $U^g \cap U = 1$ or $U$ for every $g \in G$, and the centralizer of every non-trivial element of $U$ is contained in $U$?

Let $G$ be a finite group with trivial soluble radical in which there are Sylow 2-subgroups having non-trivial intersection. Suppose that, for any two Sylow 2-subgroups $P$ and $Q$ of $G$ with $P \cap Q \neq 1$, the index $|P : P \cap Q|$ does not exceed $2^n$. Is it then true that $|P| \leqslant 2^{2n}$?

Suppose that $K$ and $L$ are distinct conjugacy classes of involutions in a finite group $G$ and $\langle x, y \rangle$ is a 2-group for all $x \in K$ and $y \in L$. Does it follow that $G \neq [K, L]$?

An intersection of some Sylow 2-subgroups is called a Sylow intersection and an intersection of a pair of Sylow 2-subgroups is called a paired Sylow intersection.
$\qquad$ a) Describe the finite groups all of whose 2-local subgroups have odd indices.
$\qquad$ b) Describe the finite groups all of whose normalizers of Sylow intersections have odd indices.
$\qquad$ c) Describe the finite groups all of whose normalizers of paired Sylow intersections have odd indices.
$\qquad$ d) Describe the finite groups in which for any two Sylow 2-subgroups $P$ and $Q$, the intersection $P \cap Q$ is normal in some Sylow 2-subgroup of $\langle P, Q \rangle$.

Do there exist finitely presented residually finite groups with recursive, but not primitive recursive, solution of the word problem?

Is every countable locally linear group embeddable in a finitely presented group?

If the finite group $G$ has the form $G = AB$ where $A$ and $B$ are nilpotent of classes $\alpha$ and $\beta$, respectively, then $G$ is soluble. Is $G^{(\alpha+\beta)} = 1$? One can ask the same question for infinite groups (or Lie algebras), but there is nothing known beyond Ito’s theorem: $A' = B' = 1$ implies $G^{(2)} = 1$.

Let $G$ be an infinite locally finite simple group. Is the centralizer of every element of $G$ infinite?

a) Let $G$ be an infinite locally finite simple group satisfying the minimum condition for 2-subgroups. Is $G = \text{PSL}_2(F)$, $F$ some locally finite field of odd characteristic, if the centralizer of every involution of $G$ is almost locally soluble?
b) Can one characterize the simple locally finite groups with the min-2 condition containing a maximal radical non-trivial 2-subgroup of rank $\leqslant 2$ as linear groups of small rank?

Is the elementary theory of lattices of $l$-ideals of lattice-ordered abelian groups decidable?

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

Does there exist a version of the Higman embedding theorem in which the degree of unsolvability of the conjugacy problem is preserved?

Is it true that a free lattice-ordered group of the variety of lattice-ordered groups defined by the law $x^{-1} |y| x \ll |y|^2$ (or, equivalently, by the law $|[x, y]| \ll |x|$), is residually linearly ordered nilpotent?

Is it true that a free lattice-ordered group of the variety of the lattice-ordered groups which are residually linearly ordered, is residually soluble linearly ordered?

Prove that the factor-group of any soluble linearly ordered group by its derived subgroup is non-periodic.

Let $G$ be a finite $p$-group with the minimal number of generators $d$, and let $r_1$ (respectively, $r_2$) be the minimal number of defining relations on $d$ generators in the sense of representing $G$ as a factor-group of a free discrete group (pro-$p$-group). It is well known that always $r_2 > d^2/4$. For each prime number $p$ denote by $c(p)$ the exact upper bound for the numbers $b(p)$ with the property that $r_2 \geqslant b(p)d^2$ for all finite $p$-groups.
$\qquad$ a) It is obvious that $r_1 \geqslant r_2$. Find a $p$-group with $r_1 > r_2$.
$\qquad$ b) Conjecture: $\lim\limits_{p \to \infty} c(p) = 1/4$.

Prove that if $G$ is a torsion-free pro-$p$-group with a single defining relation, then $\operatorname{cd} G = 2$.

Let $G$ be a group and $H$ a torsion-free subgroup of $G$ such that the augmentation ideal $I_G$ of the integral group-ring $\mathbb{Z}G$ can be decomposed as $I_G = I_H\mathbb{Z}G \oplus M$ for some $\mathbb{Z}G$-submodule $M$. Prove that $G$ is a free product of the form $G = H \ast K$.

Consider a group $G$ given by a presentation with $m$ generators and $n$ defining relations, where $m \geqslant n$. Do some $m - n$ of the given generators generate a free subgroup of $G$?

(Well-known problem). Suppose that $G$ is a finite soluble group, $A \leqslant \operatorname{Aut} G$, $C_G(A) = 1$, the orders of $G$ and $A$ are coprime, and let $|A|$ be the product of $n$ not necessarily distinct prime numbers. Is the nilpotent length of $G$ bounded above by $n$?

Let $p$ be a prime, $C$ a conjugacy class of $p$-elements of a finite group $G$ and suppose that for any two elements $x$ and $y$ of $C$ the product $xy^{-1}$ is a $p$-element. Is the subgroup generated by the class $C$ a $p$-group?

(Y. Ihara). Consider the quaternion algebra $Q$ with norm $f = x^2 - \tau y^2 - \rho z^2 + \rho \tau u^2$, $\rho, \tau \in \mathbb{Z}$. Assume that $f$ is indefinite and of $\mathbb{Q}$-rank 0, i. e. $f = 0$ for $x, y, z, u \in \mathbb{Q}$ implies $x = y = z = u = 0$. Consider $Q$ as the algebra of the matrices
$$X = \begin{pmatrix} x + \sqrt{\tau}y & \rho(z + \sqrt{\tau}u) \\ z - \sqrt{\tau}u & x - \sqrt{\tau}y \end{pmatrix}$$ with $x, y, z, u \in \mathbb{Q}$. Let $p$ be a prime, $p \nmid \rho \tau$. Consider the group $G$ of all $X$ with $x, y, z, u \in \mathbb{Z}^{(p)}$, $\operatorname{det} X = 1$, where $\mathbb{Z}^{(p)} = \{m/p^t \mid m, t \in \mathbb{Z}\}$.

Conjecture: $G$ has the congruence subgroup property, i. e. every non-central normal subgroup $N$ of $G$ contains a full congruence subgroup $N(\mathfrak{a}) = \{X \in G \mid X \equiv E \pmod{\mathfrak{a}}\}$ for some $\mathfrak{a}$. Notice that the congruence subgroup property is independent of the matrix representation of $Q$.

Let $\mathfrak{o}$ be a commutative ring with identity in which 2 is invertible and which is not generated by zero divisors. Do there exist non-standard automorphisms of $\text{GL}_n(\mathfrak{o})$ for $n \geqslant 3$?

Let $V$ be a vector space of dimension $n$ over a field. A subgroup $G$ of $GL_n(V)$ is said to be rich in transvections if $n \geqslant 2$ and for every hyperplane $H \subseteq V$ and every line $L \subseteq H$ there is at least one transvection in $G$ with residual line $L$ and fixed space $H$. Describe the automorphisms of the subgroups of $GL_2(V)$ which are rich in transvections.

What profinite groups satisfy the maximum condition for closed subgroups?

Is it the case that if $A, B$ are finitely generated soluble Hopfian groups then $A \times B$ is Hopfian?

Prove that every countable group can act faithfully as a group of automorphisms of a finitely generated soluble group (of derived length at most 4). For background to this problem, in particular its relationship with problem 8.50, see (P. M. Neumann, in: Groups–Korea, Pusan, 1988 (Lect. Notes Math., 1398), Springer, Berlin, 1989, 124–139).

Let $G$ be a countable group acting on a set $\Omega$. Suppose that $G$ is $k$-fold transitive for every finite $k$, and $G$ contains no non-trivial permutations of finite support. Is it true that $\Omega$ can be identified with the rational line $\mathbb{Q}$ in such a way that $G$ becomes a group of autohomeomorphisms?

Does every non-trivial finite group, which is free in some variety, contain a non-trivial abelian normal subgroup?

Does the free group of rank 2 have an infinite ascending chain of verbal subgroups each being generated as a verbal subgroup by a single element?

Does there exist a soluble variety of groups that is not generated by its finite groups?

A union $\mathfrak{A} = \bigcup_{\alpha} \mathfrak{V}_\alpha$ of varieties of groups (in the lattice of varieties) is called irreducible if $\bigcup_{\beta \neq \alpha} \mathfrak{V}_\beta \neq \mathfrak{A}$ for each index $\alpha$. Is every variety an irreducible union of (finitely or infinitely many) varieties each of which cannot be decomposed into a union of two proper subvarieties?

Is every recursively presented soluble group embeddable in a group finitely presented in the variety of all soluble groups of derived length $n$, for suitable $n$?

Is every countable abelian group embeddable in the center of some finitely presented group?

Suppose that $G$ and $H$ are finitely generated residually finite groups having the same set of finite homomorphic images. Are $G$ and $H$ isomorphic if one of them is a free (free soluble) group?

Let $A_{m,n}$ be the group of all automorphisms of the free soluble group of derived length $n$ and rank $m$. Is it true that
$\qquad$ a) $A_{m,n}$ is finitely generated for any $m$ and $n$?
$\qquad$ b) every automorphism in $A_{m,n}$ is induced by some automorphism in $A_{m,n+1}$?

Is there a finite group whose set of quasi-laws does not have an independent basis?

Does a non-abelian free group have an independent basis for its quasi-laws?

It is not hard to show that a finite perfect group is the normal closure of a single element. Is the same true for infinite finitely generated groups?

(P. Scott). Let $p, q, r$ be distinct prime numbers. Prove that the free product $G = C_p \ast C_q \ast C_r$ of cyclics of orders $p, q, r$ is not the normal closure of a single element.

Let $p, q, r$ be distinct primes and $u(x, y, z)$ a commutator word in three variables. Prove that there exist (infinitely many?) natural numbers $n$ such that the alternating group $\mathbb{A}_n$ can be generated by three elements $\xi, \eta, \zeta$ satisfying $\xi^p = \eta^q = \zeta^r = \xi\eta\zeta \cdot u(\xi, \eta, \zeta) = 1$.

Find a finite $p$-group that cannot be embedded in a finite $p$-group with trivial multiplicator. Notice that every finite group can be embedded in a group with trivial multiplicator.

a) Let $p$ be a prime greater than 3. Is it true that every finite group of exponent $p$ can be embedded in the commutator subgroup of a finite group of exponent $p$?
b) Does there exist a locally nilpotent group of prime exponent that coincides with its derived subgroup (and hence has no maximal subgroups)?

Let $G$ be a locally finite group which is the product of a $p$-subgroup and a $q$-subgroup, where $p$ and $q$ are distinct primes. Is $G$ a $\{p, q\}$-group?

Is an arbitrary soluble group that satisfies the minimum condition for normal subgroups countable?

(Well-known problem). Does an arbitrary uncountable locally finite group have only one end?

Is a group locally finite if it contains infinite subgroups?

Prove that a finite group is not simple if it contains a non-trivial solvable normal subgroup.

Suppose that a finite group $G$ is the product of two subgroups $A$ and $B$, where $A$ is abelian and $B$ is nilpotent. Find the dependence of the derived length of $G$ on the nilpotency class of $B$ and the order of its derived subgroup.

Is the class of all finite groups that have Hall $\pi$-subgroups closed under taking finite subdirect products?

Is a periodic residually finite group finite if it satisfies the weak minimum condition for subgroups?

Let $G$ be a finitely-presented group, and assume that $G$ has polynomial growth in the sense of Milnor. Show that $G$ has soluble word problem.

Is every lattice isomorphism between torsion-free groups having no non-trivial cyclic normal subgroups induced by a group isomorphism?