A subgroup $H$ of an arbitrary group $G$ is said to be C-closed if $H = C^2(H) = C(C(H))$, and weakly C-closed if $C^2(x) \leqslant H$ for any element $x$ of $H$. The structure of the finite groups all of whose proper subgroups are $C$-closed was studied by Gaschütz. Describe the structure of the locally finite groups all of whose proper subgroups are weakly $C$-closed.

The totality of $C$-closed subgroups of an arbitrary group $G$ is a complete lattice with respect to the operations $A \wedge B = A \cap B$, $A \vee B = C(C(A) \cap C(B))$. Describe the groups whose lattice of $C$-closed subgroups is a sublattice of the lattice of all subgroups.

A group $G$ is called of type $(FP)_\infty$ if the trivial $G$-module $\mathbb{Z}$ has a resolution by finitely generated projective $G$-modules. The class of all groups of type $(FP)_\infty$ has a couple of excellent closure properties with respect to extensions and amalgamated products (R. Bieri, Homological dimension of discrete groups, Queen Mary College Math. Notes, London, 1976). Is every periodic group of type $(FP)_\infty$ finite? This is related to the question whether there is an infinite periodic group with a finite presentation.

Is it true that every torsion-free group of type $(FP)_\infty$ (see 6.3) has finite cohomological dimension?

Is it the case that every soluble group $G$ of type $(FP)_\infty$ (see 6.3) is constructible in the sense of (G. Baumslag, R. Bieri, Math. Z., 151, no. 3 (1976), 249–257)? In other words, can $G$ be built up from the trivial group in terms of finite extensions and $HNN$-extensions?

Let $P, Q$ be permutation representations of a finite group $G$ with the same character. Suppose $P(G)$ is a primitive permutation group. Is $Q(G)$ necessarily primitive?

Suppose that $P$ is a finite 2-group. Does there exist a characteristic subgroup $L(P)$ of $P$ such that $L(P)$ is normal in $H$ for every finite group $H$ that satisfies the following conditions:
$\qquad$ 1) $P$ is a Sylow 2-subgroup of $H$,
$\qquad$ 2) $H$ is $S_4$-free, and
$\qquad$ 3) $C_H(O_2(H)) \leqslant O_2(H)$?

Can a residually finite locally normal group be embedded in a Cartesian product of finite groups in such a way that each element of the group has at most finitely many central projections?

Is the derived subgroup of any locally normal group decomposable into a product of at most countable subgroups which commute elementwise?

Let $p$ be a prime and $n$ be an integer with $p > 2n + 1$. Let $x, y$ be $p$-elements in $GL_n(\mathbb{C})$. If the subgroup $\langle x, y \rangle$ is finite, then it is an abelian $p$-group (W. Feit, J. G. Thompson, Pacif. J. Math., 11, no. 4 (1961), 1257–1262). What can be said about $\langle x, y \rangle$ in the case it is an infinite group?

Let $\mathcal{L}$ be the class of locally compact groups with no small subgroups (see D. Montgomery, L. Zippin, Topological transformation groups, New York, 1955; V. M. Glushkov, Uspekhi Matem. Nauk, 12, no. 2 (1957), 3–41 (Russian)). Study extensions of groups in this class with the objective of giving a direct proof of the following: For each $G \in \mathcal{L}$ there exists an $H \in \mathcal{L}$ and a (continuous) homomorphism $\vartheta : G \to H$ with a discrete kernel and an image $\operatorname{Im} \vartheta$ satisfying $\operatorname{Im} \vartheta \cap Z(H) = 1$ ($Z(H)$ is the center of $H$).

This result follows from the Gleason–Montgomery–Zippin solution of Hilbert’s 5th Problem (since $\mathcal{L}$ is the class of finite-dimensional Lie groups and the latter are locally linear). On the other hand a direct proof of this result would give a substantially shorter proof of the 5th Problem since the adjoint representation of $H$ is faithful on $\operatorname{Im} \vartheta$.

a) Is a metabelian group countable if it satisfies the weak minimum condition for normal subgroups?
b) Is such a group minimax if it is torsion-free?

Is it true that if a non-abelian Sylow 2-subgroup of a finite group $G$ has a non-trivial abelian direct factor, then $G$ is not simple?

Are the following lattices locally finite: the lattice of all locally finite varieties of groups? the lattice of all varieties of groups?

A variety is said to be pro-locally-finite if it is not locally finite while all of its proper subvarieties are locally finite. An example — the variety of abelian groups. How many pro-locally-finite varieties of groups are there?

A variety is called sparse if it has at most countably many subvarieties. How many sparse varieties of groups are there?

Is every variety of groups generated by its finitely generated groups that have soluble word problem?

(Well-known problem). Suppose that a class $\mathfrak{K}$ of 2-generator groups generates the variety of all groups. Is a non-cyclic free group residually in $\mathfrak{K}$?

Let $R$ be a nilpotent associative ring. Are the following two statements for a subgroup $H$ of the adjoint group of $R$ always equivalent: 1) $H$ is a normal subgroup; 2) $H$ is an ideal of the groupoid $R$ with respect to Lie multiplication?

Does there exist a supersoluble group of odd order, all of whose automorphisms are inner?

G. Higman proved that, for any prime number $p$, there exists a natural number $\chi(p)$ such that the nilpotency class of any finite group $G$ having an automorphism of order $p$ without non-trivial fixed points does not exceed $\chi(p)$. At the same time he showed that $\chi(p) \geqslant (p^2 - 1)/4$ for any such Higman’s function $\chi$. Find the best possible Higman’s function. Is it the function defined by equalities $\chi(p) = (p^2 - 1)/4$ for $p > 2$ and $\chi(2) = 1$? This is known to be true for $p \leqslant 7$.

Construct a braid that belongs to the derived subgroup of the braid group but is not a commutator.

A braid $K$ of the braid group $B_{n+1}$ is said to be smooth if removing any of the threads in $K$ transforms $K$ into a braid that is equal to 1 in $B_n$. It is known that smooth braids form a free subgroup. Describe generators of this subgroup.

The membership problem for the braid group on four strings.

(Well-known problem). Find an algorithm for calculating the rank of coefficient-free equations in a free group. The rank of an equation is the maximal rank of the free subgroup generated by a solution of this equation.

Let $D$ be a normal set of involutions in a finite group $G$ and let $\Gamma(D)$ be the graph with vertex set $D$ and edge set $\{(a, b) \mid a, b \in D, \ ab = ba \neq 1\}$. Describe the finite groups $G$ with non-connected graph $\Gamma(D)$.

Let $A$ be an elementary abelian $2$-group which is a TI-subgroup of a finite group $G$. Investigate the structure of $G$ under the hypothesis that the weak closure of $A$ in a Sylow $2$-subgroup of $G$ is abelian.

Suppose that a finite group $A$ is isomorphic to the group of all topological automorphisms of a locally compact group $G$. Does there always exist a discrete group whose automorphism group is isomorphic to $A$? This is true if $A$ is cyclic; the condition of local compactness of $G$ is essential (R. J. Wille, Indag. Math., 25, no. 2 (1963), 218–224).

Let $G$ be a residually finite Hopfian group, and let $\widehat{G}$ be its profinite completion. Is $\widehat{G}$ necessarily Hopfian (in topological sense)?

a) Suppose that $G$ is a finitely-generated residually-finite group, $d(G)$ the minimal number of generators of $G$, and $\delta(G)$ the minimal number of topological generators of the profinite completion of $G$. Is $\delta(G) = d(G)$ always true?
b) Let $G$ be a finitely generated residually finite group, $d(G)$ the minimal number of its generators, and $\delta(G)$ the minimal number of topological generators of the profinite completion of $G$. It is known that there exist groups $G$ for which $d(G) > \delta(G)$ (G. A. Noskov, Math. Notes, 33, no. 4 (1983), 249–254). Is the function $d$ bounded on the set of groups $G$ with the fixed value of $\delta(G) \geqslant 2$?

Let $F_n$ be the free profinite group of finite rank $n > 1$. Is it true that for each normal subgroup $N$ of the free profinite group of countable rank, there exists a normal subgroup of $F_n$ isomorphic to $N$?

Let $\mathfrak{o}$ be an associative ring with identity. A system of its ideals $\mathfrak{A} = \{\mathfrak{A}_{ij} \mid i, j \in \mathbb{Z}\}$ is called a carpet of ideals if $\mathfrak{A}_{ik}\mathfrak{A}_{kj} \subseteq \mathfrak{A}_{ij}$ for all $i, j, k \in \mathbb{Z}$. If $\mathfrak{o}$ is commutative, then the set $\Gamma_n(\mathcal{A}) = \{x \in \text{SL}_n(\mathfrak{o}) \mid x_{ij} \equiv \delta_{ij} \pmod{\mathfrak{A}_{ij}}\}$ is a group, the (special) congruenz-subgroup modulo the carpet $\mathfrak{A}$ (the “carpet subgroup”). Under quite general conditions, it was proved in (Yu. I. Merzlyakov, Algebra i Logika, 3, no. 4 (1964), 49–59 (Russian); see also M. I. Kargapolov, Yu. I. Merzlyakov, Fundamentals of the Theory of Groups, 3rd Ed., Moscow, Nauka, 1982, p. 145 (Russian)) that in the groups $\text{GL}_n$ and $\text{SL}_n$ the mutual commutator subgroup of the congruenz-subgroups modulo a carpet of ideals shifted by $k$ and $l$ steps is again the congruenz-subgroup modulo the same carpet shifted by $k + l$ steps. Prove analogous theorems a) for orthogonal groups; b) for unitary groups.

(R. Bieri, R. Strebel). Let $\mathfrak{o}$ be an associative ring with identity distinct from zero. A group $G$ is said to be almost finitely presented over $\mathfrak{o}$ if it has a presentation $G = F/R$ where $F$ is a finitely generated free group and the $\mathfrak{o}G$-module $R/[R, R]\otimes_{\mathbb{Z}} \mathfrak{o}$ is finitely generated. It is easy to see that every finitely presented group $G$ is almost finitely presented over $\mathbb{Z}$ and therefore also over an arbitrary ring $\mathfrak{o}$. Is the converse true?

(J. W. Grossman). The nilpotent-completion diagram of a group $G$ is as follows: $G/\gamma_1G \leftarrow G/\gamma_2G \leftarrow \dots$, where $\gamma_iG$ is the $i$th term of the lower central series and the arrows are natural homomorphisms. It is easy to see that every nilpotent-completion diagram $G_1 \leftarrow G_2 \leftarrow \dots$ is a $\gamma$-diagram, that is, every sequence $1 \to \gamma_s G_{s+1} \to G_{s+1} \to G_s \to 1$, $s = 1, 2, \dots$, is exact. Do $\gamma$-diagrams exist that are not nilpotent-completion diagrams?

(H. Wielandt, O. H. Kegel). Is a finite group $G$ soluble if it has soluble subgroups $A, B, C$ such that $G = AB = AC = BC$?

a) Let $k$ be a (commutative) field. Find all irreducible subgroups $G$ of $\text{GL}_n(k)$ having the property that $G \cap C \neq \varnothing$ for every conjugacy class $C$ of $\text{GL}_n(k)$. I conjecture that $G = \text{GL}_n(k)$ except in case $n = \text{char}\,k = 2$, the field $k$ is quadratically closed, and $G$ is conjugate to the group of all matrices of the form $\begin{pmatrix} \alpha & 0 \\ 0 & \beta \end{pmatrix}$, $\begin{pmatrix} 0 & \alpha \\ \beta & 0 \end{pmatrix}$ where $\alpha \neq 0$ and $\beta \neq 0$.
b) Is it true that an arbitrary subgroup of $GL_n(k)$ that intersects every conjugacy class is parabolic? How far is the same statement true for subgroups of other groups of Lie type?

A class of groups $\mathfrak{K}$ is said to be radical if it is closed under taking homomorphic images and normal subgroups and if every group generated by its normal $\mathfrak{K}$-subgroups also belongs to $\mathfrak{K}$. The question proposed relates to the topic “Radical classes and formulae of Narrow (first order) Predicate Calculus (NPC)”. One can show that the only non-trivial radical class definable by universal formulae of NPC is the class of all groups. Recently (in a letter to me), G. M. Bergman constructed a family of locally finite radical classes definable by formulae of NPC. Do there exist similar classes which are not locally finite and different from the class of all groups? In particular, does there exist a radical class of groups which is closed under taking Cartesian products, contains an infinite cyclic group, and is different from the class of all groups?

Let $H$ be a strongly 3-embedded subgroup of a finite group $G$. Suppose that $Z(H/O_{3'}(H))$ contains an element of order 3. Does $Z(G/O_{3'}(G))$ necessarily contain an element of order 3?

Does the set of quasi-identities holding in the class of all finite groups possess a basis in finitely many variables?

Construct a finitely generated infinite simple group requiring more than two generators.

Construct a characteristic subgroup $N$ of a finitely generated free group $F$ such that $F/N$ is infinite and simple. If no such exists, it would follow that $d(S^2) = d(S)$ for every infinite finitely generated simple group $S$, where $d(S)$ is the minimum number of generators. There is reason to believe that this is false.

If $G$ is $d$-generator group having no non-trivial finite homomorphic images (in particular, if $G$ is an infinite simple $d$-generator group) for some integer $d \geqslant 2$, must $G \times G$ be a $d$-generator group?

(C. D. H. Cooper). Let $G$ be a group, $v$ a group word in two variables such that the operation $x \odot y = v(x, y)$ defines the structure of a new group $G_v = \langle G, \odot \rangle$ on the set $G$. Does $G_v$ always lie in the variety generated by $G$?

Is every infinite binary finite $p$-group non-simple? Here $p$ is a prime number.

Is the minimal condition for abelian normal subgroups inherited by subgroups of finite index? This is true for the minimal condition for (all) abelian subgroups (J. S. Wilson, Math. Z., 114 (1970), 19–21).

Let $\mathfrak{F}$ be a local subformation of some formation $\mathfrak{X}$ of finite groups and let $\Omega$ be the set of all maximal homogeneous $\mathfrak{X}$-screens of $\mathfrak{F}$. Find a way of constructing the elements of $\Omega$ with the help of the maximal inner local screen of $\mathfrak{F}$. What can be said about the cardinality of $\Omega$? For definitions see (L. A. Shemetkov, Formations of finite groups, Moscow, Nauka, 1978 (Russian)).

Let $f$ be a local screen of a formation which contains all finite nilpotent groups and let $A$ be a group of automorphisms of a finite group $G$. Suppose that $A$ acts $f$-stably on the socle of $G/\Phi(G)$. Is it true that $A$ acts $f$-stably on $\Phi(G)$?

A group $G$ of the form $G = F \rtimes H$ is said to be a Frobenius group with kernel $F$ and complement $H$ if $H \cap H^g = 1$ for any $g \in G \setminus H$ and $F \setminus \{1\} = G \setminus \bigcup_{g \in G} H^g$. What can be said about the kernel and the complement of a Frobenius group? In particular, which groups can be kernels? complements?

Are there infinite finitely generated Frobenius groups?

A group $G$ of the form $G = F \rtimes H$ is said to be a Frobenius group with kernel F and complement H if $H \cap H^g = 1$ for any $g \in G \setminus H$ and $F \setminus \{1\} = G \setminus \bigcup_{g \in G} H^g$. Do there exist Frobenius $p$-groups?

Let $G = F \cdot \langle a \rangle$ be a Frobenius group with the complement $\langle a \rangle$ of prime order.
$\qquad$ a) Is $G$ locally finite if it is binary finite?
$\qquad$ b) Is the kernel $F$ locally finite if the groups $\langle a, a^g \rangle$ are finite for all $g \in G$?

A group $G$ is said to be (conjugacy, $p$-conjugacy) biprimitively finite if, for any finite subgroup $H$, any two elements of prime order (any two conjugate elements of prime order, of prime order $p$) in $N_G(H)/H$ generate a finite subgroup. Do the elements of finite order in a (conjugacy) biprimitively finite group $G$ form a subgroup (the periodic part of $G$)?

Are
$\qquad$ a) the Alëshin $p$-groups and
$\qquad$ b) the 2-generator Golod $p$-groups
conjugacy biprimitively finite groups (see 6.57)?

Prove that an arbitrary periodic (conjugacy) biprimitively finite group (see 6.57) (in particular, having no involutions) of finite rank is locally finite.

Do there exist an infinite finitely generated simple periodic (conjugacy) biprimitively finite group (see 6.57) which contains both involutions and non-trivial elements of odd order?

Is every infinite periodic (conjugacy) biprimitively finite group (see 6.57) without involutions non-simple?

Is a (conjugacy, $p$-conjugacy) biprimitively finite group (see 6.57) finite if it has a finite maximal subgroup ($p$-subgroup)?

An infinite group $G$ is called a monster of the first kind if it has elements of order $> 2$ and for any such an element $a$ and for any proper subgroup $H$ of $G$, there is an element $g$ in $G \setminus H$, such that $\langle a, a^g \rangle = G$. Classify the monsters of the first kind all of whose proper subgroups are finite.

A group $G$ is called a monster of the second kind if it has elements of order $> 2$ and if for any such element $a$ and any proper subgroup $H$ of $G$ there exists an infinite subset $\mathcal{M}_{a,H}$ consisting of conjugates of $a$ by elements of $G \setminus H$ such that $\langle a, c \rangle = G$ for all $c \in \mathcal{M}_{a,H}$. Do mixed monsters (that is, with elements of both finite and infinite orders) of the second kind exist? Do there exist torsion-free monsters of the second kind?