The free periodic groups $B(m, p)$ of prime exponent $p > 665$ are known to possess many properties similar to those of absolutely free groups (see S. I. Adian, The Burnside Problem and Identities in Groups, Springer, Berlin, 1979). Is it true that all normal subgroups of $B(m, p)$ are not free periodic groups?

Prove that the free periodic groups $B(m, n)$ of odd exponent $n \geqslant 665$ with $m \geqslant 2$ generators are non-amenable and that random walks on these groups do not have the recurrence property.

Prove that the periodic product of odd exponent $n \geqslant 665$ of non-trivial groups $F_1, \dots, F_k$ which do not contain involutions cannot be generated by less than $k$ elements. This would imply, on the basis of (S. I. Adian, Sov. Math. Doklady, 19, (1978), 910–913), the existence of $k$-generated simple groups which cannot be generated by less than $k$ elements, for any $k > 0$.

Is a finitely generated group with quadratic growth almost abelian?

We say that a group is indecomposable if any two of its proper subgroups generate a proper subgroup. Describe the indecomposable periodic metabelian groups.

Describe the infinite simple locally finite groups with a Chernikov Sylow 2-subgroup. In particular, are such groups the Chevalley groups over locally finite fields of odd characteristic?

Is the group $G = \langle a, b \mid a^9 = 1, ab = b^2 a^2 \rangle$ finite? This group contains $F(2, 9)$, the only Fibonacci group for which it is not yet known whether it is finite or infinite.

Suppose that $H$ is a normal subgroup of a group $G$, where $H$ and $G$ are subdirect products of the same $n$ groups $G_1, \dots, G_n$. Does the nilpotency class of $G/H$ increase with $n$?

Prove or disprove the conjecture of P. Cameron (Bull. London Math. Soc., 13, no. 1 (1981), 1–22): if $G$ is a finite primitive permutation group of subrank $m$, then either the rank of $G$ is bounded by a function of $m$ or the order of a point stabilizer is at most $m$. The subrank of a transitive permutation group is defined to be the maximum rank of the transitive constituents of a point stabilizer.

Find all groups with a Hall $2'$-subgroup.

Prove that if $F^*(G)$ is quasisimple and $\alpha \in \operatorname{Aut} G$, $|\alpha| = 2$, then $C_G(\alpha)$ contains an involution outside $Z(F^*(G))$, except when $F^*(G)$ has quaternion Sylow 2-subgroups.

If $H$ is a proper subgroup of a finite group $G$, is there always an element of prime-power order not conjugate to an element of $H$?

Is the number of maximal subgroups of a finite group $G$ at most $|G| - 1$?

Construct an explicit example of a finitely presented simple group with word problem not solvable by a primitive recursive function.

Is there an algorithm which decides if an arbitrary finitely presented solvable group is metabelian?

Suppose that a finite group $G$ is realized as the automorphism group of some torsion-free abelian group. Is it true that for every infinite cardinal $\mathfrak{m}$ there exist $2^{\mathfrak{m}}$ non-isomorphic torsion-free abelian groups of cardinality $\mathfrak{m}$ whose automorphism groups are isomorphic to $G$?

Does there exist an algorithm which decides, for a given list of group identities, whether the elementary theory of the variety given by this list is soluble, that is, by (A. P. Zamyatin, Algebra and Logic, 17, no. 1 (1978), 13–17), whether this variety is abelian?

We say that a group is sparse if the variety generated by it has at most countably many subvarieties. Does there exist a finitely generated sparse group that has undecidable word problem?

We associate with words in the alphabet $x, x^{-1}, y, y^{-1}, z, z^{-1}, \dots$ operations which are understood as functions in variables $x, y, z, \dots$. We say that a word $A$ is expressible in words $B_1, \dots, B_n$ on the group $G$ if $A$ can be constructed from the words $B_1, \dots, B_n$ and variables by means of finitely many substitutions of words one into another and replacements of a word by another word which is identically equal to it on $G$. A list of words is said to be functionally complete on $G$ if every word can be expressed on $G$ in words from this list. A word $B$ is called a Schaeffer word on $G$ if every word can be expressed on $G$ in $B$ (compare with A. V. Kuznetsov, Matem. Issledovaniya, Kishinëv, 6, no. 4 (1971), 75–122 (Russian)). Does there exist an algorithm which decides
$\qquad$ a) by a word $B$, whether it is a Schaeffer word on every group? Compare with the problem of describing all such words in (A. G. Kurosh, Theory of Groups, Moscow, Nauka, 1967, p. 435 (Russian)); here are examples of such words: $xy^{-1}$, $x^{-1}y^2 z$, $x^{-1}y^{-1}zx$.
$\qquad$ b) by a list of words, whether it is functionally complete on every group?
$\qquad$ c) by words $A, B_1, \dots, B_n$, whether $A$ is expressible in $B_1, \dots, B_n$ on every group? (This is a problem from A. V. Kuznetsov, ibid., p. 112.)
$\qquad$ d) the same for every finite group? (For finite groups, for example, $x^{-1}$ is expressible in $xy$.) For a fixed finite group an algorithm exists (compare with A. V. Kuznetsov, ibid., § 8).

The ordinal height of a variety of groups is, by definition, the supremum of order types (ordinals) of all well-ordered by inclusion chains of its proper subvarieties. It is clear that its cardinality is either finite or countably infinite, or equal to $\omega_1$. Is every countable ordinal number the ordinal height of some variety?

Is it true that the group $SL_n(q)$ contains, for sufficiently large $q$, a diagonal matrix which is not contained in any proper irreducible subgroup of $SL_n(q)$ with the exception of block-monomial ones?

Let $G(K)$ be the Chevalley group over a commutative ring $K$ associated with the root system $\Phi$ as defined in (R. Steinberg, Lectures on Chevalley groups, Yale Univ., New Haven, Conn., 1967). This group is generated by the root subgroups $x_r(K)$, $r \in \Phi$. We define an elementary carpet of type $\Phi$ over $K$ to be any collection of additive subgroups $\{\mathfrak{A}_r \mid r \in \Phi\}$ of $K$ satisfying the condition
$$c_{ij,rs}\mathfrak{A}_r^i\mathfrak{A}_s^j \subseteq \mathfrak{A}_{ir+js} \quad \text{for } r, s, ir + js \in \Phi, \ i > 0, \ j > 0,$$ where $c_{ij,rs}$ are constants defined by the Chevalley commutator formula and $\mathfrak{A}_r^i = \{a^i \mid a \in \mathfrak{A}_r\}$. What are necessary and sufficient conditions (in terms of the $\mathfrak{A}_r$) on the elementary carpet to ensure that the subgroup $\langle x_r(\mathfrak{A}_r) \mid r \in \Phi \rangle$ of $G(K)$ intersects with $x_r(K)$ in $x_r(\mathfrak{A}_r)$? See also 15.46.

Which finite simple groups can be generated by three involutions, two of which commute?

Let $A$ be a group of automorphisms of a finite non-abelian 2-group $G$ acting transitively on the set of involutions of $G$. Is $A$ necessarily soluble?

Such groups $G$ were divided into several classes in (F. Gross, J. Algebra, 40, no. 2 (1976), 348–353). For one of these classes a positive answer was given by E. G. Bryukhanova (Algebra and Logic, 20, no. 1 (1981), 1–12).

An elementary TI-subgroup $V$ of a finite group $G$ is said to be a subgroup of non-root type if $1 \neq N_V(V^g) \neq V$ for some $g \in G$. Describe the finite groups $G$ which contain a 2-subgroup $V$ of non-root type such that $[V, V^g] = 1$ implies that all involutions in $VV^g$ are conjugate to elements of $V$.

In many of the sporadic finite simple groups the 2-ranks of the centralizers of 3-elements are at most 2. Describe the finite groups satisfying this condition.

What varieties of groups $\mathfrak{V}$ have the following property: the group $G/\mathfrak{V}(G)$ is residually finite for any residually finite group $G$?

Is it true that every residually finite group in which every subgroup of finite index (including the group itself) is defined by a single defining relation is either free or isomorphic to the fundamental group of a compact surface?

We say that a profinite group is strictly complete if each of its subgroups of finite index is open. It is known (B. Hartley, Math. Z., 168, no. 1 (1979), 71–76) that finitely generated profinite groups having a finite series with pronilpotent factors are strictly complete. Is a profinite group strictly complete if it is
$\qquad$ a) finitely generated?
$\qquad$ b) finitely generated and prosoluble?

A variety of profinite groups is a non-empty class of profinite groups closed under taking subgroups, factor-groups, and Tikhonov products. A subvariety $\mathfrak{V}$ of the variety $\mathfrak{N}$ of all pronilpotent groups is said to be (locally) nilpotent if all (finitely generated) groups in $\mathfrak{V}$ are nilpotent.
$\qquad$ a) Is it true that any non-nilpotent subvariety of $\mathfrak{N}$ contains a non-nilpotent locally nilpotent subvariety?
$\qquad$ b) The same problem for the variety of all pro-$p$-groups (for a given prime $p$).

Let $G = \langle a, b \mid a^p = (ab)^3 = b^2 = (a^\sigma ba^{2/\sigma}b)^2 = 1 \rangle$, where $p$ is a prime, $\sigma$ is an integer not divisible by $p$. The group $PSL_2(p)$ is a factor-group of $G$ so that there is a short exact sequence $1 \to N \to G \to PSL_2(p) \to 1$. For each $p > 2$ there is $\sigma$ such that $N = 1$, for example, $\sigma = 4$. Let $N^{ab}$ denote the factor-group of $N$ by its commutator subgroup. It is known that for some $p$ there is $\sigma$ such that $N^{ab}$ is infinite (for example, for $p = 41$ one can take $\sigma^2 \equiv 2 \pmod{41}$), whereas for some other $p$ (for example, for $p = 43$) the group $N^{ab}$ is finite for every $\sigma$.
$\qquad$ a) Is the set of primes $p$ for which $N^{ab}$ is finite for every $\sigma$ infinite?
$\qquad$ b) Is there an arithmetic condition on $\sigma$ which ensures that $N^{ab}$ is finite?

Describe the (lattice of) subgroups of a given classical matrix group over a ring which contain the subgroup consisting of all matrices in that group with coefficients in some subring (see Ju. I. Merzljakov, J. Soviet Math., 1 (1973), 571–593).

(John S. Wilson). Is every linear $\overline{SI}$-group an $\overline{SN}$-group?

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).
$\qquad$ a) Is every primary $F^*$-group satisfying the minimum condition for subgroups almost abelian?
$\qquad$ b) Does every $F^*$-group satisfying the minimum condition for (abelian) subgroups possess the radicable part?

Does a normal subgroup $H$ in a finite group $G$ possess a complement in $G$ if each Sylow subgroup of $H$ is a direct factor in some Sylow subgroup of $G$?

Does the $Q$-theory of the class of all finite groups (in the sense of 2.40) coincide with the $Q$-theory of a single finitely presented group?

(Well-known problem). Suppose that, in a finite group $G$, each two elements of the same order are conjugate. Is then $|G| \leqslant 6$?

Let $G$ be a finitely generated group, and $N$ a minimal normal subgroup of $G$ which is an elementary abelian $p$-group. Is it true that either $N$ is finite or the growth function for the elements of $N$ with respect to a finite generating set of $G$ is bounded below by an exponential function?

Study the structure of the primitive permutation groups (finite and infinite), in which the stabilizer of any three pairwise distinct points is trivial. This problem is closely connected with the problem of describing those group which have a Frobenius group as one of maximal subgroups.

What are the primitive permutation groups (finite and infinite) which have a regular sub-orbit, that is, in which a point stabilizer acts faithfully and regularly on at least one of its orbits?

Describe the locally finite primitive permutation groups in which the centre of any Sylow 2-subgroup contains involutions stabilizing precisely one symbol. The case of finite groups was completely determined by D. Holt in 1978.

Let $p$ be a prime. The law $x \cdot x^\varphi \dots x^{\varphi^{p-1}} = 1$ from the definition of a splitting automorphism (see 1.10) gives rise to a variety of groups with operators $\langle \varphi \rangle$ consisting of all groups that admit a splitting automorphism of order $p$. Does the analogue of Kostrikin’s theorem hold for this variety, that is, do the locally nilpotent groups in this variety form a subvariety?

Does there exist a group of infinite special rank which can be represented as a product of two subgroups of finite special rank?

Is it true that a group which is a product of two almost abelian subgroups is almost soluble?

(B. Amberg). Does a group satisfy the minimum (respectively, maximum) condition on subgroups if it is a product of two subgroups satisfying the minimum (respectively, maximum) condition on subgroups?

A set of generators of a finitely presented group $G$ that consists of the least possible number $d(G)$ of generators is called a basis for $G$. Let $r_M(G)$ be the least number of relations necessary to define $G$ in the basis $M$, and $r(G)$ the minimum of $r_M(G)$ over all bases $M$ for $G$.
$\qquad$ a) It is known that $r_M(G) \leqslant d(G) + r(G)$ for any basis $M$. Does there exist a finitely presented group $G$ for which the inequality becomes equality for some basis $M$?

Let $G_1$, $G_2$ be any non-trivial groups.
$\qquad$ b) Is it true that $r_{M_1 \cup M_2}(G_1 \ast G_2) = r_{M_1}G_1 + r_{M_2}(G_2)$ for any bases $M_1, M_2$ of $G_1, G_2$, respectively?
$\qquad$ c) Is it true that $r(G_1 \ast G_2) = r(G_1) + r(G_2)$?

Suppose that $F$ is an absolutely free group, $R$ a normal subgroup of $F$, and let $\mathfrak{V}$ be a variety of groups. It is well-known (H. Neumann, Varieties of Groups, Springer, Berlin, 1967) that the group $F/\mathfrak{V}(R)$ is isomorphically embeddable in the $\mathfrak{V}$-verbal wreath product of a $\mathfrak{V}$-free group of the same rank as $F$ with $F/R$. Find a criterion indicating which elements of this wreath product belong to the image of this embedding. A criterion is known in the case where $\mathfrak{V}$ is the variety of all abelian groups (V. N. Remeslennikov, V. G. Sokolov, Algebra and Logic, 9, no. 5 (1970), 342–349).