A group $G$ is said to be potent if, to each $x \in G$ and each positive integer $n$ dividing the order of $x$ (we suppose $\infty$ is divisible by every positive integer), there exists a finite homomorphic image of $G$ in which the image of $x$ has order precisely $n$. Is the free product of two potent groups again potent?

Is $G = \langle a, b \mid a^l = b^m = (ab)^n = 1 \rangle$ conjugacy separable?

Suppose that a countable locally finite group $G$ contains no proper subgroups isomorphic to $G$ itself and suppose that all Sylow subgroups of $G$ are finite. Does $G$ possess a non-trivial finite normal subgroup?

It is known that if $\mathfrak{M}$ is a variety (quasivariety, pseudovariety) of groups, then the class $I\mathfrak{M}$ of all quasigroups that are isotopic to groups in $\mathfrak{M}$ is also a variety (quasivariety, pseudovariety), and $I\mathfrak{M}$ is finitely based if and only if $\mathfrak{M}$ is finitely based. Is it true that if $\mathfrak{M}$ is generated by a single finite group then $I\mathfrak{M}$ is generated by a single finite quasigroup?

A variety of groups is called primitive if each of its subquasivarieties is a variety. Describe all primitive varieties of groups. Is every primitive variety of groups locally finite?

Is it true that an independent basis of quasiidentities of any finite group is finite?

(A. M. Stëpin). Does there exist an infinite finitely generated amenable group of bounded exponent?

Does there exist a finitely generated simple group of intermediate growth?

Does there exist a finitely generated group which is not nilpotent-by-finite and whose growth function has as a majorant a function of the form $c^{\sqrt{n}}$ where $c$ is a constant greater than 1?

Do there exist finitely generated groups different from $\mathbb{Z}/2\mathbb{Z}$ which have precisely two conjugacy classes?

Is an abelian minimal normal subgroup $A$ of a group $G$ an elementary $p$-group if the factor-group $G/A$ is a soluble group of finite rank?

Is there a soluble group with the following properties: torsion-free of finite rank, not finitely generated, and having a faithful irreducible representation over a finite field?

Is a soluble torsion-free group minimax if it satisfies the weak minimum condition for normal subgroups?

Is a group locally finite if it decomposes into a product of periodic abelian subgroups which commute pairwise?

Describe, without using CFSG, all subgroups $L$ of a finite Chevalley group $G$ such that $G = PL$ for some parabolic subgroup $P$ of $G$.

(Well-known problem). The prime graph of a finite group $G$ is the graph with vertex set $\pi(G)$ and an edge joining $p$ and $q$ if and only if $G$ has an element of order $pq$. Describe all finite Chevalley groups over a field of characteristic 2 whose prime graph is not connected and describe the connected components.

Let $G$ be a locally normal residually finite group.
$\qquad$ a) Can $G$ be embedded in a direct product of finite groups when the factor group $G/[G, G]$ is a direct product of cyclic groups?
$\qquad$ b) Does there exist a normal subgroup $H$ of $G$ which is embeddable in a direct product of finite groups and is such that $G/H$ is a divisible abelian group?

Let $\mathfrak{S}_*$ be the smallest normal Fitting class. Are there Fitting classes which are maximal in $\mathfrak{S}_*$ (with respect to inclusion)?

Let $n(X)$ denote the minimum of the indices of proper subgroups of a group $X$. A subgroup $A$ of a finite group $G$ is called wide if $A$ is a maximal element by inclusion of the set $\{X \mid X$ is a proper subgroup of $G$ and $n(X) = n(G)\}$.
$\qquad$ a) Find all wide subgroups in finite projective special linear, symplectic, orthogonal, and unitary groups.
$\qquad$ b) Prove, without using CFSG, that $n(F_1) = |F_1 : 2F_2|$, where $F_1$ and $F_2$ are the Fischer simple groups and $2F_2$ is an extension of a group of order 2 by $F_2$.

Let $P$ be a maximal parabolic subgroup of the smallest index in a finite group $G$ of Lie type $E_6, E_7, E_8$, or ${}^2E_6$ and let $X$ be a subgroup such that $PX = G$. Is it true that $X = G$?

Let $G$ be a finite group, $B$ a block of characters of $G$, $D(B)$ its defect group and $k(B)$ (respectively, $k_0(B)$) the number of all irreducible complex characters (of height 0) lying in $B$. Conjectures:
$\qquad$ a) (R. Brauer) $k(B) \leqslant |D(B)|$; this has been proved for $p$-soluble groups (D. Gluck, K. Magaard, U. Riese, P. Schmid, J. Algebra, 279 (2004), 694–719);
$\qquad$ b) (J. B. Olsson) $k_0(B) \leqslant |D(B) : D(B)'|$, where $D(B)'$ is the derived subgroup of $D(B)$;
$\qquad$ c) (R. Brauer) $D(B)$ is abelian if and only if $k_0(B) = k(B)$.

(J. G. Thompson). Conjecture: every finite simple non-abelian group $G$ can be represented in the form $G = CC$, where $C$ is some conjugacy class of $G$.

Find an algorithm which recognizes, by an equation $w(x_1, \dots, x_n) = 1$ in a free group $F$ and by a list of finitely generated subgroups $H_1, \dots, H_n$ of $F$, whether there is a solution of this equation satisfying the condition $x_1 \in H_1, \dots, x_n \in H_n$.

a) Describe the finite groups of 2-local 3-rank 1 which have 3-rank at least 3.
b) Describe the finite groups of 2-local 3-rank 1 which have non-cyclic Sylow 3-subgroups.

Let $M$ be a subgroup of a finite group $G$, $A$ an abelian 2-subgroup of $M$, and suppose that $A^g$ is not contained in $M$ for some $g$ from $G$. Determine the structure of $G$ under the hypothesis that $\langle A, A^x \rangle = G$ whenever the subgroup $A^x, x \in G$, is not contained in $M$.

Suppose that a finite group $G$ is generated by a conjugacy class $D$ of involutions and let $D_i = \{d_1 \cdots d_i \mid d_1, \dots, d_i$ are different pairwise commuting elements of $D\}$. What is $G$, if $D_1, \dots, D_n$ are all its different conjugacy classes of involutions? For example, the Fischer groups $F_{22}$ and $F_{23}$ satisfy this condition with $n = 3$.

(Well-known problem). According to a classical theorem of Magnus, the word problem is soluble in 1-relator groups. Do there exist 2-relator groups with insoluble word problem?

(Well-known problem). A finite set of reductions $u_i \to v_i$ of words on a finite alphabet $\Sigma = \Sigma^{-1}$ is called a group set of reductions if $\text{length}(u_i) > \text{length}(v_i)$ or $\text{length}(u_i) = \text{length}(v_i)$ and $u_i > v_i$ in the lexicographical ordering, and every word in $\Sigma$ can be reduced to the unique reduced form which does not depend on the sequence of reductions. Do there exist group sets of reductions satisfying the condition $\text{length}(v_i) \leqslant 1$ for all $i$, which are different from 1) sets of trivial reductions $x^{-\varepsilon} x^\varepsilon \to 1$, $\varepsilon = \pm 1$, 2) multiplication tables $xy \to z$ of finite groups, and 3) their finite unions?

Let $k$ be a field of characteristic 0. According to (Yu. I. Merzlyakov, Proc. Steklov Inst. Math., 167 (1986), 263–266), the family $\operatorname{Rep}_k(G)$ of all canonical matrix representations of a $k$-powered group $G$ over $k$ may be regarded as an affine $k$-variety. Find an explicit form of equations defining this variety.

What locally compact groups satisfy the following condition: the product of any two closed subgroups is also a closed subgroup? Abelian groups with this property were described in (Yu. N. Mukhin, Math. Notes, 8 (1970), 755–760).

(F. Kümmich, H. Scheerer). If $H$ is a closed subgroup of a connected locally-compact group $G$ such that $HX = HX$ for every closed subgroup $X$ of $G$, then is $H$ normal?

(S. K. Grosser, W. N. Herfort). Does there exist an infinite compact $p$-group in which the centralizers of all elements are finite?

A topological group is said to be inductively compact if any finite set of its elements is contained in a compact subgroup. Is this property preserved under lattice isomorphisms in the class of locally compact groups?

Characterize the lattices of closed subgroups in locally compact groups.

Is a compactly generated inductively prosoluble locally compact group prosoluble?

A group is said to be compactly covered if it is the union of its compact subgroups. In a null-dimensional locally compact group, are maximal compactly covered subgroups closed?

Let $\Omega$ be a countable set and $\mathfrak{m}$ a cardinal number such that $\aleph_0 \leqslant \mathfrak{m} \leqslant 2^{\aleph_0}$ (we assume Axiom of Choice but not Continuum Hypothesis). Does there exist a permutation group $G$ on $\Omega$ that has exactly $\mathfrak{m}$ orbits on the power set $\mathcal{P}(\Omega)$?

Let $\Omega$ be a countably infinite set. Define a moiety of $\Omega$ to be a subset $\Sigma$ such that both $\Sigma$ and $\Omega \setminus \Sigma$ are infinite. Which permutation groups on $\Omega$ are transitive on moieties?

Let $\Omega$ be a countably infinite set. For $k \geqslant 2$ define a $k$-section of $\Omega$ to be a partition of $\Omega$ as union of $k$ infinite sets.
$\qquad$ a) Does there exist a transitive permutation group on $\Omega$ that is transitive on $k$-sections but intransitive on ordered $k$-sections?
$\qquad$ b) Does there exist a group that is transitive on $k$-sections but not on $(k + 1)$-sections?
$\qquad$ c) Does there exist a transitive permutation group on $\Omega$ that is transitive on $\aleph_0$-sections but which is a proper subgroup of $\operatorname{Sym}(\Omega)$?

Let $\Omega$ be a countable set and let $D$ be the set of total order relations on $\Omega$ for which $\Omega$ is order-isomorphic with $\mathbb{Q}$. Does there exist a transitive proper subgroup $G$ of $\operatorname{Sym}(\Omega)$ which is transitive on $D$?

a) The group $G$ described in the solution of problem 8.73 enables us to construct a projective plane of order 3 in which the lines are the elements of any conjugacy class of subgroups of order $2 \cdot 5 \cdot 7 \cdot 11$ together with four lines added in a natural way. In a similar way, we can construct a projective plane of order $p^n$ for any prime $p$ and any positive integer $n$. Does the resulting projective plane have the Galois property?

b) The group $G$ indicated in (N. D. Podufalov, Abstracts of the 9th All–Union Symp. on Group Theory, Moscow, 1984, 113–114 (Russian)) allows us to construct a projective plane of order 3: one can take as lines any class of subgroups of order $2 \cdot 5 \cdot 11 \cdot 17$ conjugate under $S$ and add four more lines in a natural way. In a similar way, one can construct projective planes of order $p^n$ for any prime $p$ and any natural $n$. Could this method be adapted for constructing new planes?

A topological group is said to be layer compact if the full inverse images of all of its compacts under mappings $x \mapsto x^n$, $n = 1, 2, \dots$, are compacts. Describe the locally compact locally soluble layer compact groups.

Let $a$ be a vector in $\mathbb{R}^n$ with rational coordinates and set
$$S(a) = \{ka + b \mid k \in \mathbb{Z}, \ b \in \mathbb{Z}^n\}.$$ It is obvious that $S(a)$ is a discrete subgroup in $\mathbb{R}^n$ of rank $n$. Find necessary and sufficient conditions in terms of the coordinates of $a$ for $S(a)$ to have an orthogonal basis with respect to the standard scalar product in $\mathbb{R}^n$.

Let $G$ be a locally compact group of countable weight and $L(G)$ the space of all its closed subgroups equipped with the $E$-topology. Then is $L(G)$ a $k$-space?

Is it true that every scattered compact can be homeomorphically embedded into the space of all closed non-compact subgroups with $E$-topology of a suitable locally compact group?

In a null-dimensional locally compact group, is the set of all compact elements closed?

Let $G$ be a compact group of weight $> \omega_2$. Is it true that the space of all closed subgroups of $G$ with respect to $E$-topology is non-dyadic?

Is every 4-Engel group
$\qquad$ a) without elements of order 2 and 5 necessarily soluble?
$\qquad$ b) satisfying the identity $x^5 = 1$ necessarily locally finite?
$\qquad$ c) (R. I. Grigorchuk) satisfying the identity $x^8 = 1$ necessarily locally finite?

Does there exist a finitely presented soluble group satisfying the maximum condition on normal subgroups which has insoluble word problem?

Does a finitely presented soluble group of finite rank have soluble conjugacy problem? Note: there is an algorithm to decide conjugacy to a given element of the group.

Is the isomorphism problem soluble for finitely presented soluble groups of finite rank?

If $G = HK$ is a soluble group and $H, K$ are minimax groups, is it true that $G$ is a minimax group?

Does there exist a finite $p$-group $G$ and a central augmented automorphism $\varphi$ of $\mathbb{Z}G$ such that $\varphi$, if extended to $\mathbb{Z}_p G$, the $p$-adic group ring, is not conjugation by unit in $\mathbb{Z}_p G$ followed by a homomorphism induced from a group automorphism?

Find all finite groups with the property that the tensor square of any ordinary irreducible character is multiplicity free.

The set of subformations of a given formation is a lattice with respect to operations of intersection and generation. What formations of finite groups have distributive lattices of subformations?

Can a product of non-local formations of finite groups be local?

(W. Gaschütz). Prove that the formation generated by a finite group has finite lattice of subformations.

Let $\mathfrak{F}$ and $\mathfrak{H}$ be local formations of finite groups and suppose that $\mathfrak{F}$ is not contained in $\mathfrak{H}$. Does $\mathfrak{F}$ necessarily have at least one minimal local non-$\mathfrak{H}$-subformation?

Two varieties are said to be S-equivalent if they have the same Mal’cev theory (D. M. Smirnov, Algebra and Logic, 22, no. 6 (1983), 492–501). What is the cardinality of the set of $S$-equivalent varieties of groups?

In any group $G$, the cosets of all of its normal subgroups together with the empty set form the block lattice $C(G)$ with respect to inclusion, which, for infinite $|G|$, is subdirectly irreducible and, for finite $|G| \geqslant 3$, is even simple (D. M. Smirnov, A. V. Reibol’d, Algebra and Logic, 23, no. 6 (1984), 459–470). How large is the class of such lattices? Is every finite lattice embeddable in the lattice $C(G)$ for some finite group $G$?

Is a finite group of the form $G = ABA$ soluble if $A$ is an abelian subgroup and $B$ is a cyclic subgroup?

Is it true that, in a group of the form $G = AB$, every subgroup $N$ of $A \cap B$ which is subnormal both in $A$ and in $B$ is subnormal in $G$? The answer is affirmative in the case of finite groups (H. Wielandt).

Is a locally soluble group periodic if it is a product of two periodic subgroups?

a) B. Jonsson’s Conjecture: Elementary equivalence is preserved under taking free products in the class of all groups, that is, if $\operatorname{Th}(G_1) = \operatorname{Th}(G_2)$ and $\operatorname{Th}(H_1) = \operatorname{Th}(H_2)$ for groups $G_1, G_2, H_1, H_2$, then $\operatorname{Th}(G_1 * H_1) = \operatorname{Th}(G_2 * H_2)$.

b) It may be interesting to consider also the following weakened conjecture: if $\operatorname{Th}(G_1) = \operatorname{Th}(G_2)$ and $\operatorname{Th}(H_1) = \operatorname{Th}(H_2)$ for countable groups $G_1, G_2, H_1, H_2$, then for any numerations of the groups $G_i$, $H_i$ there are $m$-reducibility $T_1 \equiv_m T_2$ and Turing reducibility $T_1 \equiv_T T_2$, where $T_i$ is the set of numbers of all theorems in $\operatorname{Th}(G_i * H_i)$. A proof of this weakened conjecture would be a good illustration of application of the reducibility theory to solving concrete mathematical problems.

(A. Tarski). Let $F_n$ be a free group of rank $n$; is it true that $\text{Th}(F_2) = \text{Th}(F_3)$?

Let $\mathfrak{V}$ be a variety of groups which is not the variety of all groups, and let $p$ be a prime. Is there a bound on the $p$-lengths of the finite $p$-soluble groups whose Sylow $p$-subgroups are in $\mathfrak{V}$?

(P. Cameron). Let $G$ be a finite primitive permutation group and suppose that the stabilizer $G_\alpha$ of a point $\alpha$ induces on some of its orbits $\Delta \neq \{\alpha\}$ a regular permutation group. Is it true that $|G_\alpha| = |\Delta|$?

Is it true that every infinite 2-transitive permutation groups with locally soluble point stabilizer has a non-trivial irreducible finite-dimensional representation over some field?

Let $G_1$ and $G_2$ be Lie groups with the following property: each $G_i$ contains a nilpotent simply connected normal Lie subgroup $B_i$ such that $G_i/B_i \cong SL_2(\mathbb{K})$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. Assume that $G_1$ and $G_2$ are contained as closed subgroups in a topological group $G$, that $G_1 \cap G_2 \geqslant B_1B_2$, and that no non-identity Lie subgroup of $B_1 \cap B_2$ is normal in $G$. Can it then be shown (perhaps by using the method of “amalgams” from the theory of finite groups) that the nilpotency class and the dimension of $B_1B_2$ is bounded?

Let $\mathfrak{F}$ be any local formation of finite soluble groups containing all finite nilpotent groups. Prove that $H^\mathfrak{F} K = K H^\mathfrak{F}$ for any two subnormal subgroups $H$ and $K$ of an arbitrary finite group $G$.

Find all local formations $\mathfrak{F}$ of finite groups such that every finite minimal non-$\mathfrak{F}$-group is either a Shmidt group (that is, a non-nilpotent finite group all of whose proper subgroups are nilpotent) or a group of prime order.

Find all local formations $\mathfrak{F}$ of finite groups such that, in every finite group, the set of $\mathfrak{F}$-subnormal subgroups forms a lattice.

We define a Golod group to be the $r$-generated, $r \geqslant 2$, subgroup $\langle 1 + x_1 + I, 1 + x_2 + I, \dots, 1 + x_r + I \rangle$ of the adjoint group $1 + F/I$ of the factor-algebra $F/I$, where $F$ is a free algebra of polynomials without constant terms in non-commuting variables $x_1, x_2, \dots, x_r$ over a field of characteristic $p > 0$, and $I$ is an ideal of $F$ such that $F/I$ is a non-nilpotent nil-algebra (see E. S. Golod, Amer. Math. Soc. Transl. (2), 48 (1965), 103–106). Prove that in Golod groups the centralizer of every element is infinite.

Note that Golod groups with infinite centre were constructed by A. V. Timofeenko (Math. Notes, 39, no. 5 (1986), 353–355); independently by other methods the same result was obtained in the 90s by L. Hammoudi.

Does there exist an infinite finitely generated residually finite binary finite group all of whose Sylow subgroups are finite? A. V. Rozhkov (Dr. of Sci. Disser., 1997) showed that such a group does exist if the condition of finiteness of the Sylow subgroups is weakened to local finiteness.

A group $U$ is called an $F_q$-group (where $q \in \pi(U)$) if, for each finite subgroup $K$ of $U$ and for any two elements $a, b$ of order $q$ in $T = N_U (K)/K$, there exists $c \in T$ such that the group $\langle a, b^c \rangle$ is finite. A group $U$ is called an $F^*$-group if each subgroup $H$ of $U$ is an $F_q$-group for every $q \in \pi(H)$ (V. P. Shunkov, 1977).

Does there exist a periodic residually finite $F^*$-group all of whose Sylow subgroups are finite and which is not binary finite?

(A. G. Kurosh). Is every group with the minimum condition countable?

Are the 2-elements of a group with the minimum condition contained in its locally finite radical?

Does there exist a simple group with the minimum condition possessing a non-trivial quasi-cyclic subgroup?

An infinite group, all of whose proper subgroups are finite, is called quasi-finite. Is it true that an element of a quasi-finite group $G$ is central if and only if it is contained in infinitely many subgroups of $G$?

Suppose that $G$ is a (periodic) $p$-conjugacy biprimitively finite group (see 6.57) which has a finite Sylow $p$-subgroup. Is it then true that all Sylow $p$-subgroups of $G$ are conjugate?

a) Is every binary finite 2-group of finite exponent locally finite?
b) The same question for $p$-groups for $p > 2$.