Characterize all groups (or at least all soluble groups) $G$ and fields $F$ such that every irreducible $FG$-module has finite dimension over the centre of its endomorphism ring.

Let $G$ be the amalgamated free product of two polycyclic groups, amalgamating a normal subgroup of each. Is $G$ isomorphic to a linear group? $G$ is residually finite (G. Baumslag, Trans. Amer. Math. Soc., 106, no. 2 (1963), 193–209) and the answer is “yes” if the amalgamated part is torsion-free, abelian (B. A. F. Wehrfritz, Proc. London Math. Soc., 27, no. 3 (1973), 402–424) and nilpotent (M. Shirvani, 1981, unpublished).

Let $m$ and $n$ be positive integers and $p$ a prime. Let $P_n(\mathbb{Z}_{p^m})$ be the group of all $n \times n$ matrices $(a_{ij})$ over the integers modulo $p^m$ such that $a_{ii} \equiv 1$ for all $i$ and $a_{ij} \equiv 0 \pmod{p}$ for all $i > j$. The group $P_n(\mathbb{Z}_{p^m})$ is a finite $p$-group. For which $m$ and $n$ is it regular?

Construct a finite nilpotent loop with no finite basis for its laws.

Prove that if $X$ is a finite group, $F$ is any field, and $M$ is a non-trivial irreducible $FX$-module then $$\frac{1}{|X|} \sum_{x \in X} \text{dim fix}(x) \leqslant \frac{1}{2} \text{dim } M.$$

(M. M. Day). Do the classes of amenable groups and elementary groups coincide?

Does there exist a non-amenable finitely presented group which has no free subgroups of rank 2?

(D. V. Anosov).
a) Does there exist a non-cyclic finitely-generated group $G$ containing an element $a$ such that every element of $G$ is conjugate to a power of $a$?
b) Does there exist a non-cyclic finitely presented group $G$ which contains an element $a$ such that each element of $G$ is conjugate to some power of $a$?

(C. Chou). We say that a group $G$ has property $P$ if for every finite subset $F$ of $G$, there is a finite subset $S \supset F$ and a subset $X \subset G$ such that $x_1S \cap x_2S$ is empty for any $x_1, x_2 \in X$, $x_1 \neq x_2$, and $G = \bigcup_{x \in X} Sx$. Does every group have property $P$?

Is the group $G = \langle a, b \mid a^n = 1, ab = b^3 a^3 \rangle$ finite or infinite for $n = 7$, $n = 9$, and $n = 15$? All other cases known. See also 7.7.

Consider the group
$$M = \langle x, y, z, t \mid [x, y] = [y, z] = [z, x] = (x, t) = (y, t) = (z, t) = 1 \rangle$$ where $[x, y] = x^{-1}y^{-1}xy$ and $(x, t) = x^{-1}t^{-1}x^{-1}txt$. The subgroup $H = \langle x, t, y \rangle$ is isomorphic to the braid group $\mathfrak{B}_4$, and is normally complemented by $N = \langle (zx^{-1})^M \rangle$. Is $N$ a free group (?) of countably infinite rank (?) on which $H$ acts faithfully by conjugation?

Let $D_0$ denote the class of finite groups of deficiency zero, i. e. having a presentation $\langle X \mid R \rangle$ with $|X| = |R|$.
$\qquad$ a) Does $D_0$ contain any 3-generator $p$-group for $p \geqslant 5$?
$\qquad$ b) Are the central factors of nilpotent $D_0$-groups 3-generated?
$\qquad$ c) Do soluble $D_0$-groups have bounded derived length?
$\qquad$ d) Which non-abelian simple groups can occur as composition factors of $D_0$-groups?

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p$ and $\mathfrak{g}$ be the Lie algebra of $G$. Is the number of orbits of nilpotent elements of $\mathfrak{g}$ under the adjoint action of $G$ finite?

a) Assume a group $G$ is existentially closed in the class $L\mathfrak{N}_p$ of all locally finite $p$-groups. Is it true that $G$ is characteristically simple? This is true for $G$ countable in $L\mathfrak{N}_p$ (Berthold Maier, Freiburg); in fact, up to isomorphism, there is only one such countable locally finite $p$-group.
b) Assume a group $G$ is existentially closed in one of the classes $L\mathfrak{N}^+$, $L\mathfrak{S}_\pi$, $L\mathfrak{S}^+$, $L\mathfrak{S}$ of, respectively, all locally nilpotent torsion-free groups, all locally soluble $\pi$-groups, all locally soluble torsion-free groups, or all locally soluble groups. Is it true that $G$ is characteristically simple? The existential closedness of $G$ is a local property, thus it seems difficult to obtain global properties of $G$ from it.

(Well-known problem). Is every $\widetilde{N}$-group a $\overline{Z}$-group? For definitions, see (M. I. Kargapolov, Yu. I. Merzlyakov, Fundamentals of the Theory of Groups, 3rd Edition, Moscow, Nauka, 1982, p. 205 (Russian)).

Is the class of all $\overline{Z}$-groups closed under taking normal subgroups?

An $\widetilde{\text{RN}}$-group is one whose every homomorphic image is an $\text{RN}$-group. Is the class of $\text{RN}$-groups closed under taking normal subgroups?

Is every countably infinite abelian group a verbal subgroup of some finitely generated (soluble) relatively free group?

Which of the following properties of (soluble) varieties of groups are equivalent to one another:
$\qquad$ 1) to satisfy the minimum condition for subvarieties;
$\qquad$ 2) to have at most countably many subvarieties;
$\qquad$ 3) to have no infinite independent system of identities?

What is the cardinality of the set of all varieties covering an abelian (nilpotent? Cross? hereditarily finitely based?) variety of groups? The question is related to 4.46 and 4.73.

If the commutator subgroup $G'$ of a relatively free group $G$ is periodic, must $G'$ have finite exponent?

If $G$ is a (non-abelian) finite group contained in a join $\mathfrak{A} \vee \mathfrak{B}$ of two varieties $\mathfrak{A}, \mathfrak{B}$ of groups, must there exist finite groups $A \in \mathfrak{A}, B \in \mathfrak{B}$ such that $G$ is a section of the direct product $A \times B$?

If the dihedral group $D$ of order 18 is a section of a direct product $A \times B$, must at least one of $A$ and $B$ have a section isomorphic to $D$?

Prove that every linearly ordered group of finite rank is soluble.

Does there exist an algorithm which recognizes by an identity whether it defines a variety of groups that has at most countably many subvarieties?

We call a variety passable if there exists an unrefinable chain of its subvarieties which is well-ordered by inclusion. For example, every variety generated by its finite groups is passable – this is an easy consequence of (H. Neumann, Varieties of groups, Berlin et al., Springer, 1967, Chapter 5). Do there exist non-passable varieties of groups?

Does the lattice of all varieties of groups possess non-trivial automorphisms?

Is the variety of groups finitely based if it is generated by a finitely based quasivariety of groups?

Do there exist locally nilpotent groups with trivial centre satisfying the weak maximal condition for normal subgroups?

Let $\mathfrak{X}$, $\mathfrak{Y}$ be Fitting classes of soluble groups which satisfy the Lockett condition, i. e. $\mathfrak{X} \cap \mathfrak{S}_* = \mathfrak{X}_*$, $\mathfrak{Y} \cap \mathfrak{S}_* = \mathfrak{Y}_*$ where $\mathfrak{S}$ denotes the Fitting class of all soluble groups and the lower star the bottom group of the Lockett section determined by the given Fitting class. Does $\mathfrak{X} \cap \mathfrak{Y}$ satisfy the Lockett condition?

Describe the finite groups in which every proper subgroup has a complement in some larger subgroup. Among these groups are, for example, $\text{PSL}_2(7)$ and all Sylow subgroups of symmetric groups.

Suppose $G$ is a finitely generated group such that, for any set $\pi$ of primes and any subgroup $H$ of $G$, if $G/\langle H^G \rangle$ is a finite $\pi$-group then $|G : H|$ is a finite $\pi$-number. Is $G$ nilpotent? This is true for finitely generated soluble groups.

Let $a, b$ be two elements of a group, $a$ having infinite order. Find a necessary and sufficient condition for $\bigcap_{n=1}^\infty \langle a^n, b \rangle = \langle b \rangle$.

Let $G$ be a finite group. Is it true that indecomposable projective $\mathbb{Z}G$-modules are finitely generated (and hence locally free)?

Determine the conjugacy classes of maximal subgroups in the sporadic simple groups
$\qquad$ a) $F'_{24}$;
$\qquad$ b) $F_2$;
$\qquad$ c) $F_1$.
See the current status in Atlas of Finite Group Representations (http://brauer.maths.qmul.ac.uk/Atlas/spor/M/).

(R. Griess).
a) Is $M_{11}$ a section in $O'N$?
b) The same question for $M_{24}$ in $F_2$; $J_1$ in $F_1$ and in $F_2$; $J_2$ in $F_{23}$, $F'_{24}$, and $F_2$.

b) Describe the irreducible subgroups of $\text{SL}_6(q)$.

Describe the finite groups generated by a conjugacy class $D$ of involutions which satisfies the following property: if $a, b \in D$ and $|ab| = 4$, then $[a, b] \in D$. This condition is satisfied, for example, in the case where $D$ is a conjugacy class of involutions of a known finite simple group such that $\langle C_D(a) \rangle$ is a 2-group for every $a \in D$.

What finite groups $G$ contain a normal set of involutions $D$ which contains a non-empty proper subset $T$ satisfying the following properties:
$\qquad$ 1) $C_D(a) \subseteq T$ for any $a \in T$;
$\qquad$ 2) if $a, b \in T$ and $ab = ba \neq 1$, then $C_D(ab) \subseteq T$?

Describe the finite groups all of whose soluble subgroups of odd indices have 2-length 1. For example, the groups $L_n(2^m)$ are known to satisfy this condition.

(F. Timmesfeld). Let $T$ be a Sylow 2-subgroup of a finite group $G$. Suppose that $\langle N(B) \mid B$ is a non-trivial characteristic subgroup of $T \rangle$ is a proper subgroup of $G$. Describe the group $G$ if $F^*(M) = O_2(M)$ for any 2-local subgroup $M$ containing $T$.

Prove or disprove that for all but finitely many primes $p$, the group
$$G_p = \langle a, b \mid a^2 = b^p = (ab)^3 = (b^r ab^{-2r} a)^2 = 1 \rangle,$$ where $r^2 + 1 \equiv 0 \pmod{p}$, is infinite. A solution of this problem would have interesting topological applications.

When it follows that a group is residually a $(\mathfrak{X}\cap\mathfrak{Y})$-group, if it is both residually a $\mathfrak{X}$-group and residually a $\mathfrak{Y}$-group?

Describe the automorphisms of the symplectic group $\text{Sp}_{2n}$ over an arbitrary commutative ring. Conjecture: they are all standard.

Do there exist finitely presented soluble groups in which the maximum condition for normal subgroups fails but all central sections are finitely generated?

If a finite group $G$ can be written as the product of two soluble subgroups of odd index, then is $G$ soluble?

Let $G$ be a $p$-group acting transitively as a permutation group on a set $\Omega$, let $F$ be a field of characteristic $p$, and regard $F\Omega$ as an $FG$-module. Then do the descending and ascending Loewy series of $F\Omega$ coincide?

At a conference in Oberwolfach in 1979 I exhibited a finitely generated soluble group $G$ (of derived length 3) and a non-zero cyclic $\mathbb{Z}G$-module $V$ such that $V \cong V \oplus V$. Can this be done with $G$ metabelian? I conjecture that it cannot. For background of this problem and for details of the construction see (P. M. Neumann, in: Groups–Korea, Pusan, 1988 (Lect. Notes Math., 1398), Springer, Berlin, 1989, 124–139).

(J. McKay). If $G$ is a finite group and $p$ a prime let $m_p(G)$ denote the number of ordinary irreducible characters of $G$ whose degree is not divisible by $p$. Let $P$ be a Sylow $p$-subgroup of $G$. Is it true that $m_p(G) = m_p(N_G(P))$?

(Well-known problem). Does there exist an infinite finitely presented periodic group? Compare with 6.3.

Let $n$ be a sufficiently large odd number.
$\qquad$ a) Describe the automorphisms of the free Burnside group $B(m, n)$ of exponent $n$ on $m$ generators.
$\qquad$ b) Is it true that every non-cyclic subgroup of $B(m, n)$ has a subgroup isomorphic to $B(2, n)$?

a) (Well-known problem). Classify metabelian varieties of groups (or show that this is, in a certain sense, a “wild” problem).
b) Describe the identities of 2-generated metabelian groups, that is, classify varieties generated by such groups.

It is easy to see that the set of all quasivarieties of groups, in each of which quasivarieties a non-trivial identity holds, is a semigroup under multiplication of quasivarieties. Is this semigroup free?

Let $X$ be a finite set and $f$ a mapping from the set of subsets of $X$ to the positive integers. Under the requirement that in a group generated by $X$, every subgroup $\langle Y \rangle$, $Y \subseteq X$, be nilpotent of class $\leqslant f(Y)$, is it true that the free group $G_f$ relative to this condition is torsion-free?

Does a locally compact locally nilpotent nonabelian group without elements of finite order contain a proper closed isolated normal subgroup?

Suppose that, in a locally compact locally nilpotent group $G$, all proper closed normal subgroups are compact. Does $G$ contain an open compact normal subgroup?

Describe the locally compact primary locally soluble groups which are covered by compact subgroups and all of whose closed abelian subgroups have finite rank.

Suppose that a locally compact group $G$ contains a subgroup that is topologically isomorphic to the additive group of the field of real numbers with natural topology. Is the space of all closed subgroups of $G$ connected in the Chabauty topology?

Describe the locally compact locally pronilpotent groups for which the space of closed normal subgroups is compact in the $E$-topology. The corresponding problem has been solved for discrete groups.

Suppose that the space of all closed subgroups of a locally compact group $G$ is $\sigma$-compact in the $E$-topology. Is it true that the set of closed non-compact subgroups of $G$ is at most countable?

Does the class of all finite groups possess an independent basis of quasiidentities?

Construct examples of residually finite groups which would separate Shunkov’s classes of groups with $(a, b)$-finiteness condition, (weakly) conjugacy biprimitively finite groups and (weakly) biprimitively finite groups (see 6.57). Can one derive such examples from Golod’s construction?

Does there exist a Golod group all of whose abelian subgroups have finite ranks? Here a Golod group means a finitely generated non-nilpotent subgroup of the adjoint group of a nil-ring.

Let $G = \langle a, b \mid r = 1 \rangle$ where $r$ is a cyclically reduced word that is not a proper power of any word in $a, b$. If $G$ is residually finite, is $G_t = \langle a, b \mid r^t = 1 \rangle$, $t > 1$, residually finite?

Is every 1-relator group with non-trivial torsion conjugacy separable?

Let $A, B$ be polycyclic-by-finite groups. Let $G = A \ast_H B$ where $H$ is cyclic. Is $G$ conjugacy separable?

Is every countable conjugacy-separable group embeddable in a 2-generator conjugacy-separable group?

Does there exist a finitely presented group $G$ which is not free or cyclic of prime order, having the property that every proper subgroup of $G$ is free?

We say that a finite group $G$ separates cyclic subgroups if, for any cyclic subgroups $A$ and $B$ of $G$, there is $g \in G$ such that $A \cap B^g = 1$. Is it true that $G$ separates cyclic subgroups if and only if $G$ has no non-trivial cyclic normal subgroups?

A subnormal subgroup $H \triangleleft\triangleleft\ G$ of a group $G$ is said to be good if and only if $\langle H, J \rangle \triangleleft\triangleleft\ G$ whenever $J \triangleleft\triangleleft \ G$. Is it true that when $H$ and $K$ are good subnormal subgroups of $G$, then $H \cap K$ is good?

(A known problem). Suppose $G$ is a finite primitive permutation group on $\Omega$, and $\alpha, \beta$ are distinct points of $\Omega$. Does there exist an element $g \in G$ such that $\alpha g = \beta$ and $g$ fixes no point of $\Omega$?

Give a realistic upper bound for the torsion-free rank of a finitely generated nilpotent group in terms of the ranks of its abelian subgroups. More precisely, for each integer $n$ let $f(n)$ be the largest integer $h$ such that there is a finitely generated nilpotent group of torsion-free rank $h$ with the property that all abelian subgroups have torsion-free rank at most $n$. It is easy to see that $f(n)$ is bounded above by $n(n+1)/2$. Describe the behavior of $f(n)$ for large $n$. Is $f(n)$ bounded below by a quadratic in $n$?

Do there exist strongly regular graphs with parameters $\lambda = 0$, $\mu = 2$ of degree $k > 10$? Such graphs are known for $k = 5$ and $k = 10$, their automorphism groups are primitive permutation groups of rank 3.

It is known that there exists a countable locally finite group that contains a copy of every other countable locally finite group. For which other classes of countable groups does a similar “largest” group exist? In particular, what about periodic locally soluble groups? periodic locally nilpotent groups?

Does there exist a countable infinite locally finite group $G$ that is complete, in the sense that $G$ has trivial centre and no outer automorphisms?

Let $G$ be a locally finite group containing a maximal subgroup which is Chernikov. Is $G$ almost soluble?

Let $G$ be a finite $p$-group admitting an automorphism $\alpha$ of prime order $q$ with $|C_G(\alpha)| \leqslant n$.
$\qquad$ a) If $p = q$, then does $G$ have a nilpotent subgroup of class at most 2 and index bounded by a function of $n$?
$\qquad$ b) If $p \neq q$, then does $G$ have a nilpotent subgroup of class bounded by a function of $q$ and index bounded by a function of $n$ (and, possibly, $q$)?

Let $\mathfrak{H} = \mathbb{C} \times \mathbb{R} = \{(z, r) \mid z \in \mathbb{C}, \ r > 0\}$ be the three-dimensional Poincaré space which admits the following action of the group $SL_2(\mathbb{C})$:
$$(z,r) \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \left( \frac{(az+b)(\overline{cz+d}) + a\bar{c}r^2}{|cz+d|^2 + |c|^2r^2}, \ \frac{r}{|cz+d|^2 + |c|^2r^2} \right).$$ Let $\mathfrak{o}$ be the ring of integers of the field $K = \mathbb{Q}(\sqrt{D})$, where $D < 0$, $\pi$ a prime such that $\pi\overline{\pi}$ is a prime in $\mathbb{Z}$ and let
$$\Gamma = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathfrak{o}) \ \middle|\ b \equiv 0 \pmod{\pi} \right\}.$$ Adding to the space $\Gamma \backslash \mathfrak{H}^3$ two vertices we get a three-dimensional compact space $\overline{\Gamma \backslash \mathfrak{H}^3}$.

Calculate
$$r(\pi) = \operatorname{dim}_\mathbb{Q} H_1(\overline{\Gamma \backslash \mathfrak{H}^3}, \mathbb{Q}) = \operatorname{dim}_\mathbb{Q}(\Gamma^{ab} \otimes \mathbb{Q}).$$

For example, if $D = -3$ then $r(\pi)$ is distinct from zero for the first time for $\pi \mid 73$ (then $r(\pi) = 1$), and if $D = -4$ then $r(\pi) = 1$ for $\pi \mid 137$.

In the notation of 8.82, for $r(\pi) > 0$, one can define via Hecke algebras a formal Dirichlet series with Euler multiplication (see G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, 1971). Does there exist an algebraic Hasse–Weil variety whose $\zeta$-function is this Dirichlet series? There are several conjectures.

We say that an automorphism $\varphi$ of the group $G$ is a pseudo-identity if, for all $x \in G$, there exists a finitely generated subgroup $K_x$ of $G$ such that $x \in K_x$ and $\varphi|_{K_x}$ is an automorphism of $K_x$. Let $G$ be generated by subgroups $H, K$ and let $G$ be locally nilpotent. Let $\varphi : G \to G$ be an endomorphism such that $\varphi|_H$ is pseudo-identity of $H$ and $\varphi|_K$ is an automorphism of $K$. Does it follow that $\varphi$ is an automorphism of $G$? It is known that $\varphi$ is a pseudo-identity of $G$ if, additionally, $\varphi|_K$ is pseudo-identity of $K$; it is also known that $\varphi$ is an automorphism if, additionally, $K$ is normal in $G$.

Construct a finite $p$-group $G$ whose Hughes subgroup $H_p(G) = \langle x \in G \mid |x| \neq p \rangle$ is non-trivial and has index $p^3$.

(Well-known problem). Suppose that all proper closed subgroups of a locally compact locally nilpotent group $G$ are compact. Is $G$ abelian if it is non-compact?

Find all hereditary local formations $\mathfrak{F}$ of finite groups satisfying the following condition: every finite minimal non-$\mathfrak{F}$-group is biprimary. A finite group is said to be biprimary if its order is divisible by precisely two distinct primes.