(Well-known problem). Describe the structure of the centralizers of unipotent elements in almost simple groups of Lie type.

Classify the simple groups that are isomorphic to the multiplicative groups of finite rings, in particular, of the group rings of finite groups over finite fields and over $\mathbb{Z}/n\mathbb{Z}, n \in \mathbb{Z}$.

(M. Aschbacher). A $p$-local subgroup $H$ in a group $G$ is said to be a superlocal if $H = N_G(O_p(H))$. Describe the superlocals in alternating groups and in groups of Lie type.

Is it true that the lattice of centralizers in a group is modular if it is a sublattice of the lattice of all subgroups? This is true for finite groups.

Let $k$ be a commutative ring and let $G$ be a torsion-free almost polycyclic group. Suppose that $P$ is a finitely generated projective module over the group ring $kG$ and $P$ contains two elements independent over $kG$. Is $P$ a free module?

Let $p$ be an odd prime. Is it true that every finite $p$-group possesses a set of generators of equal orders?

a) Find conditions for a group $G$, given by its presentation, under which the property that
$$\text{some term of the lower central series of $G$ is a free group} \qquad (*)$$ implies the residual finiteness of $G$.
b) Find conditions on the structure of a residually finite group $G$ which ensure property ($*$) for $G$.

For a finite group $X$, let $\chi_1(X)$ denote the totality of the degrees of all irreducible complex characters of $X$ with allowance for their multiplicities. Suppose that $\chi_1(G) = \chi_1(H)$ for groups $G$ and $H$. Clearly, then $|G| = |H|$.
$\qquad$ a) Is it true that $H$ is simple if $G$ is simple?
$\qquad$ b) Is it true that $H$ is soluble if $G$ is soluble?
It is known that $H$ is a Frobenius group if $G$ is a Frobenius group.

(I. I. Pyatetskiĭ-Shapiro). Does there exist a finite non-soluble group $G$ such that the set of characters induced by the trivial characters of representatives of all conjugacy classes of subgroups of $G$ is linearly independent?

(R. C. Lyndon).
a) Does there exist an algorithm that, given a group word $w(a, x)$, recognizes whether $a$ is equal to the identity element in the group $\langle a, x \mid a^n = 1, \ w(a, x) = 1 \rangle$?
b) Is it true that $a \neq 1$ in $G = \langle a, x \mid a^5 = 1, a^{x^2} = [a, a^x] \rangle$?

The well-known Baer-Suzuki theorem states that if every two conjugates of an element $a$ of a finite group $G$ generate a finite $p$-subgroup, then $a$ is contained in a normal $p$-subgroup.
$\qquad$ a) Does such a theorem hold in the class of periodic groups? The case $p = 2$ is of particular interest.
$\qquad$ b) Does such a theorem hold in the class of binary finite groups?

a) Suppose that $G$ is a simple locally finite group in which the centralizer of some element is a linear group, that is, a group admitting a faithful matrix representation over a field. Is $G$ itself a linear group?
b) The same question with replacement of the word “linear” by “finitary linear”. A group $H$ is finitary linear if it admits a faithful representation on an infinite-dimensional vector space $V$ such that the residue subspaces $V(1 - h)$ have finite dimensions for all $h \in H$.

Suppose that $G$ is a periodic group with an involution $i$ such that $i^g \cdot i$ has odd order for any $g \in G$. Is it true that the image of $i$ in $G/O(G)$ belongs to the centre of $G/O(G)$?

Is every finite simple group characterized by its Cartan matrix over an algebraically closed field of characteristic 2? In (R. Brandl, Arch. Math., 38 (1982), 322–323) it is shown that for any given finite group there exist only finitely many finite groups with the same Cartan matrix.

It is known that for each prime number $p$ there exists a series $a_1, a_2, \dots$ of words in two variables such that the finite group $G$ has abelian Sylow $p$-subgroups if and only if $a_k(G) = 1$ for almost all $k$. For $p = 2$ such a series is known explicitly (R. Brandl, J. Austral. Math. Soc., 31 (1981), 464–469). What about $p > 2$?

Let $V_r$ be the class of all finite groups $G$ satisfying a law $[x, \phantom{}_r y] = [x, \phantom{}_s y]$ for some $s = s(G) > r$. Here $[x, \phantom{}_1 y] = [x, y]$ and $[x, \phantom{}_{i+1} y] = [[x, \phantom{}_i y], y]$.
$\qquad$ a) Is there a function $f$ such that every soluble group in $V_r$ has Fitting length $< f(r)$? For $r < 3$ see (R. Brandl, Bull. Austral. Math. Soc., 28 (1983), 101–110).
$\qquad$ b) Is it true that $V_r$ contains only finitely many nonabelian simple groups? This is true for $r < 4$.

Let $G$ be a finite group and let $d = d(G)$ be the least positive integer such that $G$ satisfies a law $[x, \phantom{}_r y] = [x, \phantom{}_{r+d} y]$ (see 11.16) for some nonnegative integer $r = r(G)$.
$\qquad$ a) Let $e = 1$ if $d(G)$ is even and $e = 2$ otherwise. Is it true that the exponent of $G/F(G)$ divides $e \cdot d(G)$?
$\qquad$ b) If $G$ is a nonabelian simple group, does the exponent of $G$ divide $d(G)$?

Let $G(a, b) = \langle x, y \mid x = [x, \phantom{}_a y], \ y = [y, \phantom{}_b x] \rangle$. (see 11.16.) Is $G(a, b)$ finite?

It is easy to show that $G(1, b) = 1$ and one can show that $G(2, 2) = 1$. Nothing is known about $G(2, 3)$. If one could show that every minimal simple group is a quotient of some $G(a, b)$, then this would yield a very nice sequence of words in two variables to characterize soluble groups, see (R. Brandl, J. S. Wilson, J. Algebra, 116 (1988), 334–341).

(C. Sims). Is the $n$-th term of the lower central series of an absolutely free group the normal closure of the set of basic commutators (in some fixed free generators) of weight exactly $n$?

Suppose we have $[a, b] = [c, d]$ in an absolutely free group, where $a, b, [a, b]$ are basic commutators (in some fixed free generators). If $c$ and $d$ are arbitrary (proper) commutators, does it follow that $a = c$ and $b = d$?

Let $\mathfrak{N}_p$ denote the formation of all finite $p$-groups, for a given prime number $p$. Is it true that, for every subformation $\mathfrak{F}$ of $\mathfrak{N}_p$, there exists a variety $\mathfrak{M}$ such that $\mathfrak{F} = \mathfrak{N}_p \cap \mathfrak{M}$?

Characterize all $p$-groups, $p$ a prime, that can be faithfully represented as $n \times n$ triangular matrices over a division ring of characteristic $p$.

An automorphism $\varphi$ of a group $G$ is called a nil-automorphism if, for every $a \in G$, there exists $n$ such that $[a, \phantom{}_n \varphi] = 1$. Here $[x, \phantom{}_1 y] = [x, y]$ and $[x, \phantom{}_{i+1} y] = [[x, \phantom{}_i y], y]$. An automorphism $\varphi$ is called an e-automorphism if, for any two $\varphi$-invariant subgroups $A$ and $B$ such that $A \not\subseteq B$, there exists $a \in A \setminus B$ such that $[a, \varphi] \in B$. Is every $e$-automorphism of a group a nil-automorphism?

A Fitting class $\mathfrak{F}$ is said to be local if there exists a group function $f$ (for definition see (L. A. Shemetkov, Formations of Finite Groups, Moscow, Nauka, 1978 (Russian)) such that $f(p)$ is a Fitting class for every prime number $p$ and
$$\mathfrak{F} = \mathfrak{G}_{\pi(\mathfrak{F})} \cap \bigcap_{p \in \pi(\mathfrak{F})} f(p) \mathfrak{N}_p \mathfrak{G}_{p'}.$$ Is every hereditary Fitting class of finite groups local?

For the definition of the product of Fitting classes see (N. T. Vorob’ev, Math. Notes, 43, no. 2 (1988), 91–94).
$\qquad$ a) Does there exist a local product (different from the class of all finite groups and from the class of all finite soluble groups) of Fitting classes each of which is not local and is not a formation?
$\qquad$ b) Do there exist local Fitting classes which are decomposable into a non-trivial product of Fitting classes and in every such a decomposition all factors are non-local?

Does there exist a group which is not isomorphic to outer automorphism group of a metabelian group with trivial center?

What are the minimum numbers of generators for groups $G$ satisfying $S \leqslant G \leqslant \text{Aut}\,S$ where $S$ is a finite simple non-abelian group?

Suppose the prime graph of the finite group $G$ is disconnected. (This means that the set of prime divisors of the order of $G$ is the disjoint union of non-empty subsets $\pi$ and $\pi'$ such that $G$ contains no element of order $pq$ where $p \in \pi, \ q \in \pi'$.) Then P. A. Linnell (Proc. London Math. Soc., 47, no. 1 (1983), 83–127) has proved mod CFSG that there is a decomposition of $\mathbb{Z}G$-modules $\mathbb{Z} \oplus \mathbb{Z}G = A \oplus B$ with $A$ and $B$ non-projective. Find a proof independent of CFSG.

Let $F$ be a free group and $\mathfrak{f} = \mathbb{Z}F(F - 1)$ the augmentation ideal of the integral group ring $\mathbb{Z}F$. For any normal subgroup $R$ of $F$ define the corresponding ideal $\mathfrak{r} = \mathbb{Z}F(R - 1) = {}_\operatorname{id}(r - 1 \mid r \in R)$. One may identify, for instance, $F \cap (1 + \mathfrak{rf}) = R'$, where $F$ is naturally imbedded into $\mathbb{Z}F$ and $1 + \mathfrak{rf} = \{1 + a \mid a \in \mathfrak{rf}\}$.

Identify in an analogous way in terms of corresponding subgroups of $F$:
$\qquad$ a) $F \cap (1 + \mathfrak{r}_1\mathfrak{r}_2 \cdots \mathfrak{r}_n)$, where $R_i$ are normal subgroups of $F$, $i = 1, 2, \dots, n$;
$\qquad$ b) $F \cap (1 + \mathfrak{r}_1\mathfrak{r}_2\mathfrak{r}_3);$
$\qquad$ c) $F \cap (1 + \mathfrak{fs} + \mathfrak{f}^n)$, where $F/S$ is finitely generated nilpotent;
$\qquad$ d) $F \cap (1 + \mathfrak{fsf} + \mathfrak{f}^n)$;
$\qquad$ e) $F \cap (1 + \mathfrak{r}(k) + \mathfrak{f}^n)$, $n > k \geqslant 2$, where $\mathfrak{r}(k) = \mathfrak{rf}^{k-1} + \mathfrak{frf}^{k-2} + \dots + \mathfrak{f}^{k-1}\mathfrak{r}$;
$\qquad$ f) Is the quotient group $(F \cap (1 + \mathfrak{r} + \mathfrak{f}^n))/R \cdot \gamma_n(F)$ always abelian?

Is it true that the rank of a torsion-free soluble group is equal to the rank of any of its subgroups of finite index? The answer is affirmative for groups having a rational series (D. I. Zaitsev, in: Groups with restrictions on subgroups, Naukova dumka, Kiev, 1971, 115–130 (Russian)). We note also that every torsion-free soluble group of finite rank contains a subgroup of finite index which has a rational series.

Is a radical group polycyclic if it is a product of two polycyclic subgroups? The answer is affirmative for soluble and for hyperabelian groups (D. I. Zaitsev, Math. Notes, 24, no. 6 (1983), 843–851; B. Hartley, G. Shute, Quart. J. Math. Oxford, 35 (1984), 49–71).

(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 $f|_G = 1$?

Let $G(q)$ be a simple Chevalley group over a field of order $q$. Prove that there exists $m$ such that:
$\qquad$ a) the restriction of every non-one-dimensional complex representation of $G(q^m)$ to $G(q)$ contains all irreducible representations of $G(q)$ as composition factors.
$\qquad$ b) the restriction of every non-one-dimensional representation of $G(q^m)$ over a field of prime characteristic not dividing $q$ to $G(q)$ contains all irreducible representations of $G(q)$ as composition factors.

Describe the complex representations of quasisimple finite groups which remain irreducible after reduction modulo any prime number $q$. An important example: representations of degree $(p^k - 1)/2$ of the symplectic group $Sp(2k, p)$ where $k \in \mathbb{N}$ and $p$ is an odd prime.

Suppose that $H$ is a finite linear group over $\mathbb{C}$ and $h$ is an element of $H$ of prime order $p$ which is not contained in any abelian normal subgroup. Is it true that $h$ has at least $(p - 1)/2$ different eigenvalues?

Let $G = B(m, n)$ be the free Burnside group of rank $m$ and of odd exponent $n \gg 1$. Are the following statements true?
$\qquad$ a) Every 2-generated subgroup of $G$ is isomorphic to the Burnside $n$-product of two cyclic groups.
$\qquad$ b) Every automorphism $\varphi$ of $G$ such that $\varphi^n = 1$ and $b^\varphi \cdot b^{\varphi^2} \cdots b^{\varphi^n} = 1$ for all $b \in G$ is an inner automorphism (here $m > 1$).
$\qquad$ c) The group $G$ is Hopfian if $m < \infty$.
$\qquad$ d) All retracts of $G$ are free.
$\qquad$ e) Is it true that all zero divisors in the group ring $\mathbb{Z}G$ are trivial? which means that if $ab = 0$ then $a = a_1 c$, $b = db_1$ where $a_1, c, b_1, d \in \mathbb{Z}G$, $cd = 0$, and the set $\text{supp}\,c \cup \text{supp}\,d$ is contained in a cyclic subgroup of $G$.

a) Can the free Burnside group $B(m, n)$, for any $m$ and $n$, be given by defining relations of the form $v^n = 1$ such that for any natural divisor $d$ of $n$ distinct from $n$ the element $v^d$ is not trivial in $B(m, n)$?
b) Can the free Burnside group $B(m, n)$, for any $m$ and $n = 2^l \gg 1$, be given by defining relations of the form $v^n = 1$ such that for any natural divisor $d$ of $n$ distinct from $n$ the element $v^d$ is not trivial in $B(m, n)$?

Does there exist a finitely presented Noetherian group which is not almost polycyclic?

(Well-known problem). Does there exist a group which is not almost polycyclic and whose integral group ring is Noetherian?

Prove or disprove that a torsion-free group $G$ with the small cancellation condition $C'(\lambda)$ where $\lambda \ll 1$ necessarily has the $\mathscr{U}P$-property (and therefore $KG$ has no zero divisors).

Does there exist a torsion-free group having exactly three conjugacy classes and containing a subgroup of index 2?

For a finite group $X$, we denote by $k(X)$ the number of its conjugacy classes. Is it true that $k(AB) \leqslant k(A)k(B)$?

For a finite group $X$, we denote by $r(X)$ its sectional rank. Is it true that the sectional rank of a finite $p$-group, which is a product $AB$ of its subgroups $A$ and $B$, is bounded by some linear function of $r(A)$ and $r(B)$?

A $t$-$(v, k, \lambda)$ design $\mathscr{D} = (X, \mathscr{B})$ contains a set $X$ of $v$ points and a set $\mathscr{B}$ of $k$-element subsets of $X$ called blocks such that each $t$-element subset of $X$ is contained in $\lambda$ blocks. Prove that there are no nontrivial block-transitive 6-designs. (We have shown that there are no nontrivial block-transitive 8-designs and there are certainly some block-transitive, even flag-transitive, 5-designs.)

a) Does there exist a finite 3-group $G$ of nilpotency class 3 with the property $[a, a^\varphi] = 1$ for all $a \in G$ and all endomorphisms $\varphi$ of $G$? (See A. Caranti, J. Algebra, 97, no. 1 (1985), 1–13.)
b) Does there exist a finite $p$-group of nilpotency class greater than 2, with $\text{Aut}\,G = \text{Aut}_c\, G \cdot \text{Inn}\,G$, where $\text{Aut}_c\, G$ is the group of central automorphisms of $G$?
c) Does there exist a 2-Engel finite $p$-group $G$ of nilpotency class greater than 2 such that $\text{Aut}\,G = \text{Aut}_c\, G \cdot \text{Inn}\,G$?

Let $L_d$ be the homogeneous component of degree $d$ in a free Lie algebra $L$ of rank 2 over the field of order 2. What is the dimension of the fixed point space in $L_d$ for the automorphism of $L$ which interchanges two elements of a free generating set of $L$?

Is the commutator $[x, y, y, y, y, y, y]$ a product of fifth powers in the free group $\langle x, y \rangle$? If not, then the Burnside group $B(2, 5)$ is infinite.

B. Hartley (Proc. London Math. Soc., 35, no. 1 (1977), 55–75) constructed an example of a non-countable Artinian $\mathbb{Z}G$-module where $G$ is a metabelian group with the minimum condition for normal subgroups. It follows that there exists a non-countable soluble group (of derived length 3) satisfying Min-$n$. The following question arises in connection with this result and with the study of some classes of soluble groups with the weak minimum condition for normal subgroups. Is an Artinian $\mathbb{Z}G$-module countable if $G$ is a soluble group of finite rank (in particular, a minimax group)?

Let $A, C$ be abelian groups. If $A[n] = 0$, i. e. for $a \in A, na = 0$ implies $a = 0$, then the sequence
$$\frac{\operatorname{Hom}(C, A)}{n\operatorname{Hom}(C, A)} \rightarrowtail \operatorname{Hom}\left(\frac{C}{nC}, \frac{A}{nA}\right) \twoheadrightarrow \operatorname{Ext}(C, A)[n]$$ is exact. Given $\displaystyle f \in \operatorname{Hom}\left(C, \frac{A}{nA}\right)=\operatorname{Hom}\left(\frac{C}{nC}, \frac{A}{nA}\right)$ the corresponding extension $X_f$ is obtained as a pull-back:
$$\begin{array}{ccccc} A & \rightarrowtail & X_f & \twoheadrightarrow & C \\ \downarrow & & \downarrow & & \downarrow \\ A & \rightarrowtail & A & \twoheadrightarrow & \frac{A}{nA} \end{array}.$$ Use this scheme to classify certain extensions of $A$ by $C$. The case $nC = 0, A$ being torsion-free is interesting. Here $\operatorname{Ext}(C, A)[n] = \operatorname{Ext}(C, A)$. (See E. L. Lady, A. Mader, J. Algebra, 140 (1991), 36–64.)

Are there (large, non-trivial) classes $\mathfrak{X}$ of torsion-free abelian groups such that for $A, C \in \mathfrak{X}$ the group $\operatorname{Ext}(C, A)$ is torsion-free? It is a fact (E. L. Lady, A. Mader, J. Algebra, 140 (1991), 36–64) that two groups of such a class are nearly isomorphic if and only if they have equal $p$-ranks for all $p$.

(Well-known problem). A permutation group on a set $\Omega$ is called sharply doubly transitive if for any two pairs $(\alpha, \beta)$ and $(\gamma, \delta)$ of elements of $\Omega$ such that $\alpha \neq \beta$ and $\gamma \neq \delta$, there is exactly one element of the group taking $\alpha$ to $\gamma$ and $\beta$ to $\delta$. Does every sharply doubly transitive group possess a non-trivial abelian normal subgroup?

(P. Kleidman). Do the sporadic simple groups of Rudvalis $Ru$, Mathieu $M_{22}$, and Higman–Sims $HS$ embed into the simple group $E_7(5)$?

Is it true that in the group of coloured braids only the identity braid is a conjugate to its inverse? (For definition see 10.24.)

Is it true that extraction of roots in the group of coloured braids is uniquely determined?

a) Does every infinite residually finite group contain an infinite abelian subgroup? This is equivalent to the following: does every infinite residually finite group contain a non-identity element with an infinite centralizer?

By a famous theorem of Shunkov a torsion group with an involution having a finite centralizer is a virtually soluble group. Therefore we may assume that in our group all elements have odd order. One should start, perhaps, with the following:

b) Does every infinite residually $p$-group contain an infinite abelian subgroup?

An upper composition factor of a group $G$ is a composition factor of some finite quotient of $G$. Is there any restriction on the set of non-abelian upper composition factors of a finitely generated group?

Describe the finite groups which contain a tightly embedded subgroup $H$ such that a Sylow 2-subgroup of $H$ is a direct product of a quaternion group of order 8 and a non-trivial elementary group.

A $TI$-subgroup $A$ of a group $G$ is called a subgroup of root type if $[A, A^g] = 1$ whenever $N_A(A^g) \neq 1$. Describe the finite groups containing a cyclic subgroup of order 4 as a subgroup of root type.

Is it true that the hypothetical Moore graph with 3250 vertices of valence 57 has no automorphisms of order 2? M. Aschbacher (J. Algebra, 19, no. 4 (1971), 538–540) proved that this graph is not a graph of rank 3.

Let $F$ be a non-abelian free pro-$p$-group. Is it true that the subset $\{r \in F \mid \operatorname{cd}(F/\langle r \rangle) \leqslant 2\}$ is dense in $F$? Here $\langle r \rangle$ denotes the closed normal subgroup of $F$ generated by $r$.

Describe the groups over which any equation is soluble. In particular, is it true that this class of groups coincides with the class of torsion-free groups? It is easy to see that a group, over which every equation is soluble, is torsion-free. On the other hand, S. D. Brodskiĭ (Siberian Math. J., 25, no. 2 (1984), 235–251) showed that any equation is soluble over a locally indicable group.

Suppose that $G$ is a one-relator group containing non-trivial elements of finite order and $N$ is a subgroup of $G$ generated by all elements of finite order. Is it true that any subgroup of $G$ that intersects $N$ trivially is a free group? One can show that the answer is affirmative in the cases where $G/N$ has non-trivial centre or satisfies a non-trivial identity.

Let $\pi(G)$ denote the set of prime divisors of the order of a finite group $G$. Are there only finitely many finite simple groups $G$, different from alternating groups, which have a proper subgroup $H$ such that $\pi(H) = \pi(G)$?

Conjecture: any finitely generated soluble torsion-free pro-$p$-group with decidable elementary theory is an analytic pro-$p$-group.

(Yu. L. Ershov). Is the elementary theory of a free pro-$p$-group decidable?

Does there exist a torsion-free group with exactly 3 classes of conjugate elements such that no non-trivial conjugate class contains a pair of inverse elements?

Can every fully ordered group be embedded in a fully ordered group (continuing the given order) with only 3 classes of conjugate elements?

A group $G$ acting on a set $\Omega$ will be said to be $1$-$\{2\}$-transitive if it acts transitively on the set $\Omega^{1,\{2\}} = \{(\alpha, \{\beta, \gamma\}) \mid \alpha, \beta, \gamma$ distinct$\}$. Thus $G$ is $1$-$\{2\}$-transitive if and only if it is transitive and a stabilizer $G_\alpha$ is 2-homogeneous on $\Omega \setminus \{\alpha\}$. The problem is to classify all (infinite) permutation groups that are $1$-$\{2\}$-transitive but not 3-transitive.

Let $F$ be an infinite field or a skew-field.
$\qquad$ a) Find all transitive subgroups of $PGL(2, F)$ acting on the projective line $F \cup \{\infty\}$.
$\qquad$ b) What conditions on the subring $R$ of $F$ will ensure that $\text{PGL}(d+1, R)$ is flag-transitive on the projective $d$-space $\text{PG}(d, F)$?
$\qquad$ c) What are the flag-transitive subgroups of $PGL(d + 1, F)$?
$\qquad$ d) What subgroups of $PGL(d + 1, F)$ are 2-transitive on the points of $PG(d, F)$?

Let $A$ be a finite group with a normal subgroup $H$. A subgroup $U$ of $H$ is called an $A$-covering subgroup of $H$ if $\bigcup_{a\in A} U^a = H$. Is there a function $f : \mathbb{N} \to \mathbb{N}$ such that whenever $U < H < A$, where $A$ is a finite group, $H$ is a normal subgroup of $A$ of index $n$, and $U$ is an $A$-covering subgroup of $H$, the index $\lvert H : U \rvert \leqslant f(n)$? (We have shown that the answer is “yes” if $U$ is a maximal subgroup of $H$.)

Suppose that a variety of groups $\mathfrak{V}$ is non-regular, that is, the free group $F_{n+1}(\mathfrak{V})$ is embeddable in $F_n(\mathfrak{V})$ for some $n$. Is it true that then every countable group in $\mathfrak{V}$ is embeddable in an $n$-generated group in $\mathfrak{V}$?

If a relatively free group is finitely presented, is it virtually nilpotent?

Let $G$ be a non-elementary hyperbolic group and let $G^n$ be the subgroup generated by the $n$-th powers of the elements of $G$.
$\qquad$ a) (M. Gromov). Is it true that $G/G^n$ is infinite for some $n = n(G)$?
$\qquad$ b) Is it true that $\bigcap_{n=1}^\infty G^n = \{1\}$?

Let us consider the class of groups with $n$ generators and $m$ relators. A subclass of this class is called dense if the ratio of the number of presentations of the form $\langle a_1, \dots, a_n \mid R_1, \dots, R_m \rangle$ (where $|R_i| = d_i$) for groups from this subclass to the number of all such presentations converges to 1 when $d_1 + \dots + d_m$ tends to infinity. Prove that for every $k < m$ and for any $n$ the subclass of groups all of whose $k$-generator subgroups are free is dense.

(Well-known problem). Is the group of collineations of a finite non-Desarguesian projective plane defined over a semi-field soluble? (The hypothesis on the plane means that the corresponding regular set, see 10.48, is closed under addition.)

(Well-known problem). Describe the finite translation planes whose collineation groups act doubly transitively on the set of points of the line at infinity.

An isomorphism of groups of points of algebraic groups is called semialgebraic if it can be represented as a composition of an isomorphism of translation of the field of definition and a rational morphism.
$\qquad$ a) Is it true that the existence of an isomorphism of groups of points of two directly undecomposable algebraic groups with trivial centres over an algebraically closed field implies the existence of a semialgebraic isomorphism of the groups of points?
$\qquad$ b) Is it true that every isomorphism of groups of points of directly undecomposable algebraic groups with trivial centres defined over algebraic fields is semialgebraic? A field is called algebraic if all its elements are algebraic over the prime subfield.

Let $G$ be a finite group of automorphisms of an infinite field $F$ of characteristic $p$. Taking integral powers of the elements of $F$ and the action of $G$ define the action of the group ring $\mathbb{Z}G$ of $G$ on the multiplicative group of $F$. Is it true that any subfield of $F$ that contains the images of all elements of $F$ under the action of some fixed element of $\mathbb{Z}G \setminus p\mathbb{Z}G$ contains infinitely many $G$-invariant elements of $F$?

Let $G$ be a primitive permutation group on a finite set $\Omega$ and suppose that, for $\alpha \in \Omega$, $G_\alpha$ acts 2-transitively on one of its orbits in $\Omega \setminus \{\alpha\}$. By (C. E. Praeger, J. Austral. Math. Soc. (A), 45, 1988, 66–77) either
$\qquad$ (a) $T \leqslant G \leqslant \operatorname{Aut} T$ for some nonabelian simple group $T$, or
$\qquad$ (b) $G$ has a unique minimal normal subgroup which is regular on $\Omega$.
For what classes of simple groups in (a) is a classification feasible? Describe the examples in as explicit a manner as possible. Classify all groups in (b).

A topological group is said to be F-balanced if for any subset $X$ and any neighborhood of the identity $U$ there is a neighborhood of the identity $V$ such that $VX \subseteq XU$. Is every $F$-balanced group balanced, that is, does it have a basis of neighborhoods of the identity consisting of invariant sets?

Let $R$ be the normal closure of an element $r$ in a free group $F$ with the natural length function and suppose that $s$ is an element of minimal length in $R$. Is it true that $s$ is conjugate to one of the following elements: $r, r^{-1}, [r, f], [r^{-1}, f]$ for some $f \in F$?

Is the conjugacy problem soluble for finitely generated abelian-by-polycyclic groups?

Is the isomorphism problem soluble
$\qquad$ a) for finitely generated metabelian groups?
$\qquad$ b) for finitely generated soluble groups of finite rank?

Let $F$ be a free pro-$p$-group with a basis $X$ and let $R = r^F$ be the closed normal subgroup generated by an element $r$. We say that an element $s$ of $F$ is associated to $r$ if $s^F = R$.
$\qquad$ a) Suppose that none of the elements associated to $r$ is a $p$-th power of an element in $F$. Is it true that $F/R$ is torsion-free?
$\qquad$ b) Among the elements associated to $r$ there is one that depends on the minimal subset $X'$ of the basis $X$. Let $x \in X'$. Is it true that the images of the elements of $X \setminus \{x\}$ in $F/R$ freely generate a free pro-$p$-group?

Does every group $G = \langle x_1, \dots, x_n \mid r_1 = \dots = r_m = 1 \rangle$ possess, in a natural way, a homomorphic image $H = \langle x_1, \dots, x_n \mid s_1 = \dots = s_m = 1 \rangle$
$\qquad$ a) such that $H$ is a torsion-free group?
$\qquad$ b) such that the integral group ring of $H$ is embeddable in a skew field?

If the stronger assertion b) is true, then this will give an explicit method of finding elements $x_{i_1}, \dots, x_{i_{n-m}}$ which generate a free group in $G$. Such elements exist by N. S. Romanovskiĭ’s theorem (Algebra and Logic, 16, no. 1 (1977), 62–67).

(Well-known problem). Is the automorphism group of a free metabelian group of rank $n \geqslant 4$ finitely presented?

We define the length $l(g)$ of an Engel element $g$ of a group $G$ to be the smallest number $l$ such that $[h, g; l] = 1$ for all $h \in G$. Here $[h, g; 1] = [h, g]$ and $[h, g; i + 1] = [[h, g; i], g]$. Does there exist a polynomial function $\phi(x, y)$ such that $l(uv) \leqslant \phi(l(u), l(v))$? Up to now, it is unknown whether a product of Engel elements is again an Engel element.

Let $k$ be an infinite cardinal number. Describe the epimorphic images of the Cartesian power $\prod_k \mathbb{Z}$ of the group $\mathbb{Z}$ of integers.

Let $\mathfrak{V}$ be a variety of groups and let $k$ be an infinite cardinal number. A group $G \in \mathfrak{V}$ of rank $k$ is called almost free in $\mathfrak{V}$ if each of its subgroups of rank less than $k$ is contained in a subgroup of $G$ which is free in $\mathfrak{V}$. For what $k$ do there exist almost free but not free in $\mathfrak{V}$ groups of rank $k$?

Prove that a hereditary formation $\mathfrak{F}$ of finite soluble groups is local if every finite soluble non-simple minimal non-$\mathfrak{F}$-group is a Shmidt group (that is, a non-nilpotent finite group all of whose proper subgroups are nilpotent).

What are the soluble hereditary non-one-generator formations of finite groups all of whose proper hereditary subformations are one-generated?

Is the variety of all lattices generated by the block lattices (see 9.62) of finite groups?

Describe all simply reducible groups, that is, groups such that all their characters are real and the tensor product of any two irreducible representations contains no multiple components. This question is interesting for physicists. Is every finite simply reducible group soluble?

Suppose that $G$ is a $p$-group $G$ containing an element $a$ of order $p$ such that the subgroup $\langle a, a^g \rangle$ is finite for any $g$ and the set $C_G(a) \cap a^G$ is finite. Is it true that $G$ has non-trivial centre? This is true for 2-groups.

(a) Is it true that, for a given number $n$, there exist only finitely many finite simple groups each of which contains an involution which commutes with at most $n$ involutions of the group?
(b) Is it true that there are no infinite simple groups satisfying this condition?

Are there only finitely many finite simple groups with a given set of all different values of irreducible characters on a single element?

Let $r$ be the number of conjugacy classes of elements in a finite (simple) group $G$.
$\qquad$ a) (R. Brauer). Find the best-possible estimate of the form $\lvert G \rvert \leqslant f(r)$.
$\qquad$ b) Is it true that $|G| \leqslant \exp(r)$?

Find (in group-theoretic terms) necessary and sufficient conditions for a finite group to have complex irreducible characters having defect 0 for more than one prime number dividing the order of the group. Express the number of such characters in the same terms.

Is it true that every periodic conjugacy biprimitively finite group (see 6.57) can be obtained from 2-groups and binary finite groups by taking extensions? This question is of independent interest for $p$-groups, $p$ an odd prime.

Does there exist a Golod group (see 9.76) with finite centre?

Does there exist a residually soluble but insoluble group satisfying the maximum condition on subgroups?

Is a 2-group satisfying the minimum condition for centralizers necessarily locally finite?

Let $G$ be a finite group of order $p^a \cdot q^b \dots$, where $p, q, \dots$ are distinct primes. Introduce distinct variables $x_p, x_q, \dots$ corresponding to $p, q, \dots$ . Define functions $f, \phi$ from the lattice of subgroups of $G$ to the polynomial ring $\mathbb{Z}[x_p, x_q, \dots]$ as follows: (1) if $H$ has order $p^\alpha \cdot q^\beta \dots$, then $f(H) = x_p^\alpha \cdot x_q^\beta \dots$; (2) for all $H \leqslant G$, we have $\sum_{K \leqslant H} \phi(K) = f(H)$. Then $f(G), \phi(G)$ may be called the order and Eulerian polynomials of $G$. Substituting $p^m, q^m, \dots$ for $x_p, x_q, \dots$ in these polynomials we get the $m$th power of the order of $G$ and the number of ordered $m$-tuples of elements that generate $G$ respectively. It is known that if $G$ is $p$-solvable, then $\phi(G)$ is a product of a polynomial in $x_p$ and a polynomial in the remaining variables. Consequently, if $G$ is solvable, $\phi(G)$ is the product of a polynomial in $x_p$ by a polynomial in $x_q$ by . . . . Are the converses of these statements true
$\qquad$ a) for solvable groups?
$\qquad$ b) for $p$-solvable groups?

Its relatively free group of given rank has a presentation $F/N$, where $F$ is absolutely free of the same rank and $N$ fully invariant in $F$. The associated Lie ring $\mathscr{L}(F/N)$ has a presentation $L/J$, where $L$ is the free Lie ring of the same rank and $J$ an ideal of $L$.
$\qquad$ a) Is $J$ always fully invariant in $L$ if $\mathfrak{V}$ is any variety of groups?
$\qquad$ b) Is $J$ fully invariant in $L$ if $\mathfrak{V}$ is the Burnside variety of all groups of given exponent $q$, where $q$ is a prime-power, $q \geqslant 4$?

Can every periodic group be embedded in a simple periodic group?

Is there a non-linear locally finite simple group each of whose proper subgroups is residually finite?

Is every locally finite simple group absolutely simple? A group $G$ is said to be absolutely simple if the only composition series of $G$ is $\{1, G\}$. For equivalent formulations see (R. E. Phillips, Rend. Sem. Mat. Univ. Padova, 79 (1988), 213–220).

Is it true that the union of an ascending series of groups of Lie type of equal ranks over some fields should be also a group of Lie type of the same rank over a field? (For locally finite fields the answer is “yes”, the theorem in this case is due to V. Belyaev, A. Borovik, B. Hartley & G. Shute, and S. Thomas.)

Is it possible to embed $\text{SL}(2, \mathbb{Q})$ in the multiplicative group of some division ring?

Does every infinite locally finite simple group $G$ contain a nonabelian finite simple subgroup? Is there an infinite tower of such subgroups in $G$?

Let $L = L(K(p))$ be the associated Lie ring of a free countably generated group $K(p)$ of the Kostrikin variety of locally finite groups of a given prime exponent $p$. Is it true that
$\qquad$ a) $L$ is a relatively free Lie ring?
$\qquad$ b) all identities of $L$ follow from multilinear identities of $L$?
$\qquad$ c) all identities of $L$ follow from a finite number of identities of $L$?

(B. Hartley). Is it true that the derived length of a nilpotent periodic group admitting a regular automorphism of prime-power order $p^n$ is bounded by a function of $p$ and $n$?

Is every locally graded group of finite special rank almost hyperabelian?

Suppose that $X$ is a free non-cyclic group, $N$ is a non-trivial normal subgroup of $X$ and $T$ is a proper subgroup of $X$ containing $N$. Is it true that $[T, N] < [X, N]$ if $N$ is not maximal in $X$?

The dimension of a partially ordered set $\langle P, \leqslant \rangle$ is, by definition, the least cardinal number $\delta$ such that the relation $\leqslant$ is an intersection of $\delta$ relations of linear order on $P$. Is it true that, for any Chernikov group which does not contain a direct product of two quasicyclic groups over the same prime number, the subgroup lattice has finite dimension? Expected answer: yes.

Let $\mathfrak{X}$ be a soluble non-empty Fitting class. Is it true that every finite non-soluble group possesses an $\mathfrak{X}$-injector?

What are the hereditary soluble local formations $\mathfrak{F}$ of finite groups such that every finite group has an $\mathfrak{F}$-covering subgroup?

Is it true that, for any non-empty set of primes $\pi$, the $\pi$-length of any finite $\pi$-soluble group does not exceed the derived length of its Hall $\pi$-subgroup?

Let $\mathfrak{F}$ be a soluble saturated Fitting formation. Is it true that $l_{\mathfrak{F}}(G) \leqslant f(c_{\mathfrak{F}}(G))$, where $c_{\mathfrak{F}}(G)$ is the length of a composition series of some $\mathfrak{F}$-covering subgroup of a finite soluble group $G$? This is Problem 17 in (L. A. Shemetkov, Formations of Finite Groups, Moscow, Nauka, 1978 (Russian)). For the definition of the $\mathfrak{F}$-length $l_{\mathfrak{F}}(G)$, see ibid.

Does there exist a local formation of finite groups which has a non-trivial decomposition into a product of two formations and in any such a decomposition both factors are non-local? Local products of non-local formations do exist.

Does every non-zero submodule of a free module over the group ring of a torsion-free group contain a free cyclic submodule?

(Well-known problem). For a given group $G$, define the following sequence of groups: $A_1(G) = G, \ A_{i+1}(G) = \operatorname{Aut}(A_i(G))$. Does there exist a finite group $G$ for which this sequence contains infinitely many non-isomorphic groups?

Let $F$ be a non-cyclic free group and $R$ a non-cyclic subgroup of $F$. Is it true that if $[R, R]$ is a normal subgroup of $F$ then $R$ is also a normal subgroup of $F$?

Let $G$ be a finite group admitting a regular elementary abelian group of automorphisms $V$ of order $p^n$. Is it true that the subgroup $H = \bigcap_{v \in V \setminus \{1\}} [G, v]$ is nilpotent? In the case of an affirmative answer, does there exist a function depending only on $p$, $n$, and the derived length of $G$ which bounds the nilpotency class of $H$?

Do there exist a constant $h$ and a function $f$ with the following property: if a finite soluble group $G$ admits an automorphism $\phi$ of order 4 such that $|C_G(\phi)| \leqslant m$, then $G$ has a normal series $1 \leqslant M \leqslant N \leqslant G$ such that the index $|G : N|$ does not exceed $f(m)$, the group $N/M$ is nilpotent of class $\leqslant 2$, and the group $M$ is nilpotent of class $\leqslant h$?

Is every group of exponent 12 locally finite?

A group $G$ is said to be a $K$-group if for every subgroup $A \leqslant G$ there exists a subgroup $B \leqslant G$ such that $A \cap B = 1$ and $\langle A, B \rangle = G$. Is it true that normal subgroups of $K$-groups are also $K$-groups?