Let $U(K)$ denote the group of units of a ring $K$. Let $G$ be a finite group, $\mathbb{Z}G$ the integral group ring of $G$, and $\mathbb{Z}_p G$ the group ring of $G$ over the residues modulo a prime number $p$. Describe the homomorphism from $U(\mathbb{Z}G)$ into $U(\mathbb{Z}_p G)$ induced by reducing the coefficients modulo $p$. More precisely, find the kernel and the image of this homomorphism and an explicit transversal over the kernel.

Does there exist a finitely based variety of groups $\mathfrak{V}$ such that the word problem is solvable in $F_n(\mathfrak{V})$ for every positive integer $n$, but is unsolvable in $F_\infty(\mathfrak{V})$ (with respect to a free generating system)?

Let $\mathfrak{M}$ be an arbitrary variety of groups. Is it true that every infinitely generated projective group in $\mathfrak{M}$ is an $\mathfrak{M}$-free product of countably generated projective groups in $\mathfrak{M}$?

Let $G$ be a group with a normal (pro-) 2-subgroup $N$ such that $G/N$ is isomorphic to $GL_n(2)$, inducing its natural module on $N/\Phi(N)$, the Frattini factor-group of $N$. For $n = 3$ determine $G$ such that $N$ is as large as possible. (For $n > 3$ it can be proved that already the Frattini subgroup $\Phi(N)$ of $N$ is trivial; for $n = 2$ there exists $G$ such that $N$ is the free pro-2-group generated by two elements).

Groups $A$ and $B$ are said to be locally equivalent if for every finitely generated subgroup $X \leqslant A$ there is a subgroup $Y \leqslant B$ isomorphic to $X$, and conversely, for every finitely generated subgroup $Y \leqslant B$ there is a subgroup $X \leqslant A$, isomorphic to $Y$. We call a group $G$ categorical if $G$ is isomorphic to any group that is locally equivalent to $G$. Is it true that a periodic locally soluble group $G$ is categorical if and only if $G$ is hyperfinite with Chernikov Sylow subgroups? (A group is hyperfinite if it has a well ordered ascending normal series with finite factors.)

(B. Hartley). Is it true that a locally finite group containing an element with Chernikov centralizer is almost soluble?

(B. Hartley). Is it true that a simple locally finite group containing a finite subgroup with finite centralizer is linear?

(B. Hartley). Is it true that a locally soluble periodic group has a finite normal series with locally nilpotent factors if it contains an element
$\qquad$ a) with finite centralizer?
$\qquad$ b) with Chernikov centralizer?

Is it true that a locally soluble periodic group containing a finite nilpotent subgroup with Chernikov centralizer has a finite normal series with locally nilpotent factors?

Is there a function $f : \mathbb{N} \to \mathbb{N}$ such that, for every soluble group $G$ of derived length $k$ generated by a set $A$, the validity of the identity $x^4 = 1$ on each subgroup generated by at most $f(k)$ elements of $A$ implies that $G$ is a group of exponent 4?

Is a torsion-free group almost polycyclic if it has a finite set of generators $a_1, \dots, a_n$ such that every element of the group has a unique presentation in the form $a_1^{k_1} \dots a_n^{k_n}$, where $k_1, \dots, k_n \in \mathbb{Z}$?

Is the group of all automorphisms of an arbitrary hyperbolic group finitely presented?

Let $G$ and $H$ be finite $p$-groups with isomorphic Burnside rings. Is the nilpotency class of $H$ bounded by some function of the class of $G$? There is an example where $G$ is of class 2 and $H$ is of class 3.

Is the lattice of quasivarieties of nilpotent torsion-free groups of nilpotency class $\leqslant 2$ distributive?

Does a free non-abelian nilpotent group of class 3 possess an independent basis of quasiidentities in the class of torsion-free groups?

Is every locally nilpotent group with minimum condition on centralizers hypercentral?

A representation of a group $G$ on a vector space $V$ is called a nil-representation if for every $v$ in $V$ and $g$ in $G$ there exists $n = n(v, g)$ such that $v(g - 1)^n = 0$. Is it true that in zero characteristic every irreducible nil-representation is trivial? (In prime characteristic it is not true.)

Let $F$ be a finitely generated non-Abelian free group and let $G$ be the Cartesian (unrestricted) product of countable infinity of copies of $F$. Must the Abelianization $G/G'$ of $G$ be torsion-free?

Suppose that both a group $Q$ and its normal subgroup $H$ are subgroups of the direct product $G_1 \times \dots \times G_n$ such that for each $i$ the projections of both $Q$ and $H$ onto $G_i$ coincide with $G_i$. If $Q/H$ is a $p$-group, is $Q/H$ a regular $p$-group?

Is it true that the generating series of the growth function of every finitely generated group with one defining relation represents an algebraic (or even a rational) function?

a) Is there an infinite finitely generated residually finite $p$-group, in which the order $|g|$ of an arbitrary element $g$ does not exceed $f(\delta(g))$, where $\delta(g)$ is the length of $g$ with respect to a fixed set of generators and $f(n)$ is a function growing at $n \to \infty$ slower than any power function $n^\lambda, \lambda > 0$?
b) What is the minimal possible rate of growth of the function $\pi(n) = \max_{\delta(g)\leqslant n} |g|$ for the class of groups indicated in part a) of this problem? It is known (R. I. Grigorchuk, Math. USSR–Izv., 25 (1985), 259–300) that there exist $p$-groups with $\pi(n) \leqslant n^\lambda$ for some $\lambda > 0$. At the same time, it follows from a result of E. I. Zel’manov that $\pi(n)$ is not bounded if $G$ is infinite.

Let the group $G = AB$ be the product of two polycyclic subgroups $A$ and $B$, and assume that $G$ has an ascending normal series with locally nilpotent factors (i. e. $G$ is radical). Is it true that $G$ is polycyclic?

Let the group $G$ have a finite normal series with infinite cyclic factors (containing of course $G$ and $\{1\}$). Is it true that $G$ has a non-trivial outer automorphism?

Let $G$ be a non-discrete topological group with only finitely many ultrafilters that converge to the identity. Is it true that $G$ contains a countable open subgroup of exponent 2?

Let $(G, \tau)$ be a topological group with finite semigroup $\tau(G)$ of ultrafilters converging to the identity (I. V. Protasov, Siberian Math. J., 34, no. 5 (1993), 938–951). Is it true that $\tau(G)$ is a semigroup of idempotents?

Is it true that a countable topological group of exponent 2 with unique free ultrafilter converging to the identity has a basis of neighborhoods of the identity consisting of subgroups?

(B. Amberg). Suppose that $G = AB = AC = BC$ for a group $G$ and its subgroups $A, B, C$.
$\qquad$ a) Is $G$ a Chernikov group if $A, B, C$ are Chernikov groups?
$\qquad$ b) Is $G$ almost polycyclic if $A, B, C$ are almost polycyclic?

(D. M. Evans). A permutation group on an infinite set is cofinitary if its non-identity elements fix only finitely many points. Is it true that a closed cofinitary permutation group is locally compact (in the topology of pointwise convergence)?

Given an infinite set $\Omega$, define an algebra $A$ (the reduced incidence algebra of finite subsets) as follows. Let $V_n$ be the set of functions from the set of $n$-element subsets of $\Omega$ to the rationals $\mathbb{Q}$. Now let $A = \bigoplus V_n$, with multiplication as follows: for $f \in V_n$, $g \in V_m$, and $|X| = m + n$, let $(fg)(X) = \sum f(Y)g(X \setminus Y)$, where the sum is over the $n$-element subsets $Y$ of $X$. If $G$ is a permutation group on $\Omega$, let $A^G$ be the algebra of $G$-invariants in $A$.

Conjecture: If $G$ has no finite orbits on $\Omega$, then $A^G$ is an integral domain.

A group $G$ is called a B-group if every primitive permutation group which contains the regular representation of $G$ is doubly transitive. Are there any countable B-groups?

Let $G$ be a permutation group on a set $\Omega$. A sequence of points of $\Omega$ is a base for $G$ if its pointwise stabilizer in $G$ is the identity. The greedy algorithm for a base chooses each point in the sequence from an orbit of maximum size of the stabilizer of its predecessors. Is it true that there is a universal constant $c$ with the property that, for any finite primitive permutation group, the greedy algorithm produces a base whose size is at most $c$ times the minimal base size?

(Well-known problem). On a group, a partial order that is directed upwards and has linearly ordered cone of positive elements is said to be semilinear if this order is invariant under right multiplication by the group elements. Can every semilinear order of a group be extended to a right order of the group?

(F. Gross). Is a normal subgroup of a finite $D_\pi$-group (see 3.62) always a $D_\pi$-group?

(I. D. Macdonald). If the identity $[x, y]^n = 1$ holds on a group, is the derived subgroup of the group periodic?

Does every non-soluble pro-$p$-group of cohomological dimension 2 contain a free non-abelian pro-$p$-subgroup?

For a finitely generated pro-$p$-group $G$ set $a_n(G) = \dim_{\mathbb{F}_p} I^n/I^{n+1}$, where $I$ is the augmentation ideal of the group ring $\mathbb{F}_p[[G]]$. We define the growth of $G$ to be the growth of the sequence $\{a_n(G)\}_{n\in\mathbb{N}}$.
$\qquad$ a) If the growth of $G$ is exponential, does it follow that $G$ contains a free pro-$p$-subgroup of rank 2?
$\qquad$ b) (A. Lubotzky, A. Shalev). Is the growth of $G$ exponential if $G$ contains a finitely generated closed subgroup of exponential growth?
$\qquad$ c) Do there exist pro-$p$-groups of finite cohomological dimension which are not $p$-adic analytic, and whose growth is slower than an exponential one?

Let $G$ be a torsion-free pro-$p$-group, $U$ an open subgroup of $G$. Suppose that $U$ is a pro-$p$-group with a single defining relation. Is it true that then $G$ is also a pro-$p$-group with a single defining relation?

Let $G = F/R$ be a pro-$p$-group with one defining relation, where $R$ is the normal subgroup of a free pro-$p$-group $F$ generated by a single element $r \in F^p[F, F]$.
$\qquad$ a) Suppose that $r = t^p$ for some $t \in F$; can $G$ contain a Demushkin group as a subgroup?
$\qquad$ b) Do there exist two pro-$p$-groups $G_1 \supset G_2$ with one defining relation, where $G_1$ has elements of finite order, while the subgroup $G_2$ is torsion-free?

Let $A$ be an associative ring with unity and with torsion-free additive group, and let $F^A$ be the tensor product of a free group $F$ by $A$ (A. G. Myasnikov, V. N. Remeslennikov, Siberian Math. J., 35, no. 5 (1994), 986–996); then $F^A$ is a free exponential group over $A$; in (A. G. Myasnikov, V. N. Remeslennikov, Int. J. Algebra Comput., 6 (1996), 687–711), it is shown how to construct $F^A$ in terms of free products with amalgamation.
$\qquad$ a) (G. Baumslag). Is $F^A$ residually nilpotent torsion-free?
$\qquad$ b) Is $F^A$ a linear group?
$\qquad$ c) (G. Baumslag). Is the Magnus homework of $F^\mathbb{Q}$ into the group of power series over the rational number field $\mathbb{Q}$ faithful or not?
$\qquad$ d) Is the universal theory of $F^A$ decidable?
$\qquad$ e) (G. Baumslag). Can free $A$-groups be characterized by a length function?
$\qquad$ f) (G. Baumslag). Does a free $\mathbb{Q}$-group admit a free action on some $\Lambda$-tree? See definition in (R. Alperin, H. Bass, in: Combinatorial group theory and topology, Alta, Utah, 1984 (Ann. Math. Stud., 111), Princeton Univ. Press, 1987, 265–378).

A group $G$ is said to be $\omega$-residually free if, for every finite set of non-trivial elements of $G$, there is a homomorphism of $G$ into a free group such that the images of all these elements are non-trivial. Is every finitely-generated $\omega$-residually free group embeddable in a free $\mathbb{Z}[x]$-group?

Is the elementary theory of the class of all groups acting freely on $\Lambda$-trees decidable?

Prove that the tensor $A$-completion (see A. G. Myasnikov, V. N. Remeslenni kov, Siberian Math. J., 35, no. 5 (1994), 986–996) of a free nilpotent group can be non-nilpotent.

(G. R. Robinson). Let $G$ be a finite group and $B$ be a $p$-block of characters of $G$. Conjecture: If the defect group $D = D(B)$ of the block $B$ is non-abelian, and if $|D : Z(D)| = p^a$, then each character in $B$ has height strictly less than $a$.

For any partition of an arbitrary group $G$ into finitely many subsets $G = A_1 \cup \dots \cup A_n$, there exists a subset of the partition $A_i$ and a finite subset $F \subseteq G$, such that $G = A_i^{-1}A_i F$ (I. V. Protasov, Siberian Math. J., 34, no. 5 (1993), 938–952). Can the subset $F$ always be chosen so that $|F| \leqslant n$? This is true for amenable groups.

Every infinite group $G$ of regular cardinality $\mathfrak{m}$ can be partitioned into two subsets $G = A_1 \cup A_2$ so that $A_1 F \neq G$ and $A_2 F \neq G$ for every subset $F \subset G$ of cardinality less than $\mathfrak{m}$. Is this statement true for groups of singular cardinality?

Can every uncountable abelian group of finite odd exponent be partitioned into two subsets so that neither of them contains cosets of infinite subgroups? Among countable abelian groups, such partitions exist for groups with finitely many involutions.

Can every countable abelian group with finitely many involutions be partitioned into two subsets that are dense in every group topology?

(W. W. Comfort, J. van Mill). A topological group is said to be irresolvable if every two of its dense subsets intersect non-trivially. Does every non-discrete irresolvable group contain an infinite subgroup of exponent 2?

(V. I. Malykhin). Can a topological group be partitioned into two dense subsets, if there are infinitely many free ultrafilters on the group converging to the identity?

Let $\mathfrak{F}$ be a local Fitting class. Is it true that there are no maximal elements in the partially ordered by inclusion set of the Fitting classes contained in $\mathfrak{F}$ and distinct from $\mathfrak{F}$?

Is every finite modular lattice embeddable in the lattice of formations of finite groups?

The dimension of a finitely based variety of algebras $\mathfrak{V}$ is defined to be the maximal length of a basis (that is, an independent generating set) of the $SC$-theory $SC(\mathfrak{V})$, which consists of the strong Mal’cev conditions satisfied on $\mathfrak{V}$. The dimension is defined to be infinite if the lengths of bases in $SC(\mathfrak{V})$ are not bounded. Does every finite abelian group generate a variety of finite dimension?

Let $a, b$ be elements of finite order of the infinite group $G = \langle a, b \rangle$. Is it true that there are infinitely many elements $g \in G$ such that the subgroup $\langle a, b^g \rangle$ is infinite?

a) Is it true that, for $p$ sufficiently large, every (finite) $p$-group can be a complement in some Frobenius group (see 6.55)?
b) Is it true that every group is embeddable in the kernel of some Frobenius group (see 6.53)?

Does there exist a Golod group (see 9.76), which is isomorphic to an AT-group? For a definition of an AT-group see (A. V. Rozhkov, Math. Notes, 40, no. 5 (1986), 827–836).

(A. Shalev). Let $G$ be a finite $p$-group of sectional rank $r$, and $\varphi$ an automorphism of $G$ having exactly $m$ fixed points. Is the derived length of $G$ bounded by a function depending on $r$ and $m$ only?

Let $\varphi$ be an automorphism of prime order $p$ of a finite group $G$ such that $C_G(\varphi) \leqslant Z(G)$.
$\qquad$ a) Is $G$ soluble if $p = 3$? V. D. Mazurov and T. L. Nedorezov proved in (Algebra and Logic, 35, no. 6 (1996), 392–397) that the group $G$ is soluble for $p = 2$, and there are examples of unsoluble $G$ for all $p > 3$.
$\qquad$ b) If $G$ is soluble, is the derived length of $G$ bounded in terms of $p$?
$\qquad$ c) (V. K. Kharchenko). If $G$ is a $p$-group, is the derived length of $G$ bounded in terms of $p$? (V. V. Bludov produced a simple example showing that the nilpotency class cannot be bounded.)

Let $\varphi$ be an automorphism of prime order $p$ of a nilpotent (periodic) group $G$ such that $C_G(\varphi)$ is a group of finite sectional rank $r$. Does $G$ possess a normal subgroup $N$ which is nilpotent of class bounded by a function of $p$ only and is such that $G/N$ is a group of finite sectional rank bounded in terms of $r$ and $p$?

One can show that any extension of shape $\mathbb{Z}^d \cdot SL_d(\mathbb{Z})$ is residually finite, unless possibly $d = 3$ or $d = 5$. Are there in fact any extensions of this shape that fail to be residually finite when $d = 5$? As shown in (P. R. Hewitt, Groups/St. Andrews '93 in Galway, Vol. 2 (London Math. Soc. Lecture Note Ser., 212), Cambridge Univ. Press, 1995, 305–313), there is an extension of $\mathbb{Z}^3$ by $SL_3(\mathbb{Z})$ that is not residually finite. Usually it is true that an extension of an arithmetic subgroup of a Chevalley group over a rational module is residually finite. Is it ever false, apart from the examples of shape $\mathbb{Z}^3 \cdot SL_3(\mathbb{Z})$?

If a locally graded group $G$ is a product of two subgroups of finite special rank, is $G$ of finite special rank itself?

We call a metric space narrow if it is quasiisometric to a subset of the real line, and wide otherwise. Let $G$ be a group with the finite set of generators $A$, and let $\Gamma = \Gamma(G, A)$ be the Cayley graph of $G$ with respect to the natural metric. Suppose that, after deleting any narrow subset $L$ from $\Gamma$, at most two connected components of the graph $\Gamma \setminus L$ can be wide, and there exists at least one such a subset $L$ yielding exactly two wide components in $\Gamma \setminus L$. Is it true that $\Gamma$ is quasiisometric to an Euclidean or a hyperbolic plane?

Let $U$ and $V$ be non-cyclic subgroups of a free group. Does the inclusion $[U, U] \leqslant [V, V]$ imply that $U \leqslant V$?

Let $\pi_e(G)$ denote the set of orders of elements of a group $G$. For $\Gamma \subseteq \mathbb{N}$ let $h(\Gamma)$ denote the number of non-isomorphic finite groups $G$ with $\pi_e(G) = \Gamma$. Is there a number $k$ such that, for every $\Gamma$, either $h(\Gamma) \leqslant k$, or $h(\Gamma) = \infty$?

Let $\pi_e(G)$ denote the set of orders of elements of a group $G$. A group $G$ is said to be an $OC_n$-group if $\pi_e(G) = \{1, 2, \dots, n\}$. Is every $OC_n$-group locally finite? Do there exist infinite $OC_n$-groups for $n \geqslant 7$?

A finite simple group is called a $K_n$-group if its order is divisible by exactly $n$ different primes. The number of $K_3$-groups is known to be 8. The $K_4$-groups are classified mod CFSG (W. J. Shi, in: Group Theory in China (Math. Appl., 365), Kluwer, 1996, 163–181) and some significant further results are obtained in (Yann Bugeaud, Zhenfu Cao, M. Mignotte, J. Algebra, 241 (2001), 658–668). But the question remains: is the number of $K_4$-groups finite or infinite?

Let $F$ be a
$\qquad$ a) free;
$\qquad$ b) free metabelian
group of finite rank. Let $M$ denote the set of all endomorphisms of $F$ with non-cyclic images. Can one choose two elements $g, h \in F$ such that, for every $\varphi, \psi \in M$, equalities $\varphi(g) = \psi(g)$ and $\varphi(h) = \psi(h)$ imply that $\varphi = \psi$, that is, the endomorphisms in $M$ are uniquely determined by their values at $g$ and $h$?

We call a group $G$ containing an involution $i$ a $T_0$-group if
$\qquad$ 1) the order of the product of any two involutions conjugate to $i$ is finite;
$\qquad$ 2) all 2-subgroups of $G$ are either cyclic or generalized quaternion;
$\qquad$ 3) the centralizer $C$ of the involution $i$ in $G$ is infinite, distinct from $G$, and has finite periodic part;
$\qquad$ 4) the normalizer of any non-trivial $i$-invariant finite subgroup in $G$ either is contained in $C$ or has periodic part which is a Frobenius group (see 6.55) with abelian kernel and finite complement of even order;
$\qquad$ 5) for every element $c$ not contained in $C$ for which $ci$ is an involution there is an element $s$ of $C$ such that $\langle c, c^s \rangle$ is an infinite subgroup.

Let $G$ be a $T_0$-group, $i$ an involution in $G$ and $G = \langle i^g \mid g \in G \rangle$. Is the centralizer $C_G(i)$ residually periodic?