H. Bass (Topology, 4, no. 4 (1966), 391–400) has constructed explicitly a proper subgroup of finite index in the group of units of the integer group ring of a finite cyclic group.
$\qquad$ a) Calculate the index of Bass’ subgroup.
$\qquad$ b) (A. A. Bovdi). Construct analogous subgroups for finite abelian non-cyclic groups.

Is it true that every non-discrete group topology (in an abelian group) can be strengthened up to a maximal complete group topology?

A $p$-group is called thin if every set of pairwise incomparable (by inclusion) normal subgroups contains $\leqslant p + 1$ elements. Is the number of thin pro-$p$-groups finite?

Let $G$ be a group and assume that we have $[x, y]^3 = 1$ for all $x, y \in G$. Is $G'$ of finite exponent? Is $G$ soluble? By a result of N. D. Gupta and N. S. Mendelssohn, 1967, we know that $G'$ is a 3-group.

Does there exist a countable non-trivial filter in the lattice of quasivarieties of metabelian torsion-free groups?

Let $G$ be a finitely generated group and suppose that $H$ is a $p$-subgroup of $G$ such that $H$ contains no non-trivial normal subgroups of $G$ and $HX = XH$ for any subgroup $X$ of $G$. Is then $G/C_G(H^G)$ a $p$-group where $H^G$ is the normal closure of $H$?

Is it true that every radical hereditary formation of finite groups is a composition one?

Let $\mathfrak{V}$ be a non-trivial variety of groups and let $a_1, \dots, a_r$ freely generate a free group $F_r(\mathfrak{V})$ in $\mathfrak{V}$. We say $F_r(\mathfrak{V})$ strongly discriminates $\mathfrak{V}$ just in case every finite system of inequalities $w_i(a_1, \dots, a_r, x_1, \dots, x_k) \neq 1$ for $1 \leqslant i \leqslant n$ having a solution in some $F_s(\mathfrak{V})$ containing $F_r(\mathfrak{V})$ as a varietally free factor in the sense of $\mathfrak{V}$, already has a solution in $F_r(\mathfrak{V})$. Does there exist $\mathfrak{V}$ such that for some integer $r > 0$, $F_r(\mathfrak{V})$ discriminates but does not strongly discriminate $\mathfrak{V}$? What about the analogous question for general algebras in the context of universal algebra?

Following Bass, call a group tree-free if there is an ordered Abelian group $\Lambda$ and a $\Lambda$-tree $X$ such that $G$ acts freely without inversion on $X$.
$\qquad$ a) Must every finitely generated tree-free group satisfy the maximal condition for Abelian subgroups?
$\qquad$ b) The same question for finitely presented tree-free groups.

(P. Neumann). Can the free group on two generators be embedded in $\text{Sym}(\mathbb{N})$ so that the image of every non-identity element has only a finite number of orbits?

Suppose that $G, H$ are countable or finite groups and $A$ is a proper subgroup of $G$ and $H$ containing no non-trivial subgroup normal in both $G$ and $H$. Can $G * _A H$ be embedded in $\operatorname{Sym}(\mathbb{N})$ so that the image is highly transitive?

Is the conjugacy problem for nilpotent finitely generated lattice-ordered groups soluble?

If $\langle \Omega, \leqslant \rangle$ is the countable universal poset, then $G = \operatorname{Aut}(\langle \Omega, \leqslant \rangle)$ is simple (A. M. W. Glass, S. H. McCleary, M. Rubin, Math. Z., 214, no. 1 (1993), 55–66). If $H$ is a subgroup of $G$ such that $\lvert G : H \rvert < 2^{\aleph_0}$ and $H$ is transitive on $\Omega$, does $H = G$?

If $T$ is a countable theory, does there exist a model $\mathcal{A}$ of $T$ such that the theory of $\text{Aut}(\mathcal{A})$ is undecidable?

Suppose that, in a finite 2-group $G$, any two elements are conjugate whenever their normal closures coincide. Is it true that the derived subgroup of $G$ is abelian?

Is the class of groups of recursive automorphisms of arbitrary models closed with respect to taking free products?

Find a description of autostable periodic abelian groups.

Is it true that, for every $n \geqslant 2$ and every two epimorphisms $\varphi$ and $\psi$ of a free group $F_{2n}$ of rank $2n$ onto $F_n \times F_n$, there exists an automorphism $\alpha$ of $F_{2n}$ such that $\alpha\varphi = \psi$?

(Well-known problem). Is R. Thompson’s group
$$\begin{align} F &{}= \langle x_0, x_1, \dots \mid x_n^{x_i} = x_{n+1}, \ i < n, \ n = 1, 2, \dots \rangle \\
&{}= \langle x_0, \dots, x_4 \mid x_1^{x_0} = x_2, \ x_2^{x_0} = x_3, \ x_2^{x_1} = x_3, \ x_3^{x_1} = x_4, \ x_3^{x_2} = x_4 \rangle \end{align}$$ amenable?

Let $R$ be a commutative ring with identity and let $G$ be a finite group. Prove that the ring $a(RG)$ of $R$-representations of $G$ has no non-trivial idempotents.

Let $\Delta(G)$ be the augmentation ideal of the integer group ring of an arbitrary group $G$. Then $D_n(G) = G \cap (1 + \Delta^n(G))$ contains the $nth$ lower central subgroup $\gamma_n(G)$ of $G$.
$\qquad$ a) Is it true that $D_n(G)/\gamma_n(G)$ is central in $G/\gamma_n(G)$?
$\qquad$ b) Is it true that $D_n(G)/\gamma_n(G)$ has exponent dividing 2?

The index of permutability of a group $G$ is defined to be the minimal integer $k \geqslant 2$ such that for each $k$-tuple $x_1, \dots, x_k$ of elements in $G$ there is a non-identity permutation $\sigma$ on $k$ symbols such that $x_1 \cdots x_k = x_{\sigma(1)} \cdots x_{\sigma(k)}$. Determine the index of permutability of the symmetric group $S_n$.

Given a ring $R$ with identity, the automorphisms of $R[[x]]$ sending $x$ to $x(1 + \sum_{i=1}^\infty a_i x^i)$, $a_i \in R$, form a group $N(R)$. We know that $N(\mathbb{Z})$ contains a copy of the free group $F_2$ of rank 2 and, from work of A. Weiss, that $N(\mathbb{Z}/p\mathbb{Z})$ contains a copy of every finite $p$-group (but not of $\mathbb{Z}_{p^\infty}$), $p$ a prime. Does $N(\mathbb{Z}/p\mathbb{Z})$ contain a copy of $F_2$?

Let $G$ be a finite group acting irreducibly on a vector space $V$. An orbit $\alpha^G$ for $\alpha \in V$ is said to be $p$-regular if the stabilizer of $\alpha$ in $G$ is a $p'$-subgroup. Does $G$ have a regular orbit on $V$ if it has a $p$-regular orbit for every prime $p$?

(Shi Shengming). Is it true that a finite $p$-soluble group $G$ has a $p$-block of defect zero if and only if there exists an element $x \in O_{p'}(G)$ such that $C_G(x)$ is a $p'$-subgroup?

Let $G$ be a simple locally finite group. We say that $G$ is a group of finite type if there is a non-trivial permutational representation of $G$ such that some finite subgroup of $G$ has no regular orbits. Investigate and, perhaps, classify the simple locally finite groups of finite type. Partial results see in (B. Hartley, A. Zalesski, Isr. J. Math., 82 (1993), 299–327, J. London Math. Soc., 55 (1997), 210–230; F. Leinen, O. Puglisi, Illinois J. Math., 47 (2003), 345–360).

Let $G$ be a group. A function $f : G \to \mathbb{C}$ is called
$\qquad$ 1) normed if $f(1) = 1$;
$\qquad$ 2) central if $f(gh) = f(hg)$ for all $g, h \in G$;
$\qquad$ 3) positive-definite if $\sum_{k, l} f(g_k^{-1}g_l)\bar{c}_k c_l \geqslant 0$ for any $g_1, \dots, g_n \in G$ and any $c_1, \dots, c_n \in \mathbb{C}$.

Classify the infinite simple locally finite groups $G$ which possess functions satisfying 1)–3). The simple Chevalley groups are known to have no such functions, while such functions exist on locally matrix (or stable) classical groups over finite fields. The question is motivated by the theory of $C^*$-algebras, see § 9 in (A. M. Vershik, S. V. Kerov, J. Sov. Math., 38 (1987), 1701–1733).

Classify the locally finite groups for which the augmentation ideal of the complex group algebra is a simple ring. The problem goes back to I. Kaplansky (1965).

(O. N. Golovin). On the class of all groups, do there exist associative operations which satisfy the postulates of MacLane and Mal’cev (that is, which are free functorial and hereditary) and which are different from taking free and direct products?

For relatively free groups $G$, prove or disprove the following conjecture of P. Hall: if a word $v$ takes only finitely many values on $G$ then the verbal subgroup $vG$ is finite.

Prove an analogue of Higman’s theorem for the Burnside variety $\mathfrak{B}_n$ of groups of odd exponent $n \gg 1$, that is, prove that every recursively presented group of exponent $n$ can be embedded in a finitely presented (in $\mathfrak{B}_n$) group of exponent $n$.

Suppose that $G$ is a finite group and $x$ is an element of $G$ such that the subgroup $\langle x, y \rangle$ has odd order for any $y$ conjugate to $x$ in $G$. Prove, without using CFSG, that the normal closure of $x$ in $G$ is a group of odd order.

Describe the finite groups $G$ such that the sum of the cubes of the degrees of all irreducible complex characters is at most $\lvert G \rvert \cdot \log_2 \lvert G \rvert$. The question is interesting for applications in the theory of signal processing.

Suppose that $\mathfrak{F}$ is a radical composition formation of finite groups. Prove that $\langle H, K \rangle^\mathfrak{F} = \langle H^\mathfrak{F}, K^\mathfrak{F} \rangle$ for every finite group $G$ and any subnormal subgroups $H$ and $K$ of $G$.

Let $p$ be a prime, $V$ an $n$-dimensional vector space over the field of $p$ elements, and let $G$ be a subgroup of $\text{GL}(V)$. Let $S = S[V^*]$ be the symmetric algebra on $V^*$, the dual of $V$. Let $T = S^G$ be the ring of invariants and let $b_m$ be the dimension of the homogeneous component of degree $m$. Then the Poincaré series $\sum_{m \geqslant 0} b_m t^m$ is a rational function with a Laurent power series expansion $\sum_{i \geqslant -n} a_i (1 - t)^i$ about $t = 1$ where $a_{-n} = 1/|G|$.

Conjecture: $a_{-n+1} = r / (2|G|)$ where $r = \sum_W ((p-1)\alpha_W + s_W - 1)$, the sum is taken over all maximal subspaces $W$ of $V$, and $\alpha_W$, $s_W$ are defined by $|G_W| = p^{\alpha_W} \cdot s_W$ where $p \nmid s_W$ and $G_W$ denotes the pointwise stabilizer of $W$.

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

(J. G. Thompson). For a finite group $G$, we denote by $N(G)$ the set of all orders of the conjugacy classes of $G$. Is it true that if $G$ is a finite non-abelian simple group, $H$ a finite group with trivial centre and $N(G) = N(H)$, then $G$ and $H$ are isomorphic?

(W. J. Shi). Must a finite group and a finite simple group be isomorphic if they have equal orders and the same set of orders of elements?

Let $\varphi$ be an irreducible $p$-modular character of a finite group $G$. Find the best-possible estimate of the form $\varphi(1)_p \leqslant f(\lvert G \rvert_p)$. Here $n_p$ is the $p$-part of a positive integer $n$.

Let $F$ be a free group on two generators $x, y$ and let $\varphi$ be the automorphism of $F$ defined by $x \mapsto y$, $y \mapsto xy$. Let $G$ be a semidirect product of $F/(F''(F')^2)$ and $\langle \varphi \rangle$. Then $G$ is just-non-polycyclic. What is the cohomological dimension of $G$ over $\mathbb{Q}$? (It is either 3 or 4.)

Describe the automorphisms of the Sylow $p$-subgroup of a Chevalley group of normal type over $\mathbb{Z}/p^m\mathbb{Z}$, $m \geqslant 2$, where $p$ is a prime.

(Well-known problem). Does there exist an infinite finitely-generated residually finite $p$-group such that each subgroup is either finite or of finite index?

(P. Hall). Is there a non-trivial group which is isomorphic with every proper extension of itself by itself?

(P. Hall). Must a non-trivial group, which is isomorphic to each of its non-trivial normal subgroups, be either free of infinite rank, simple, or infinite cyclic? (Lennox, Smith and Wiegold, 1992, have shown that a finitely generated group of this kind which has a proper normal subgroup of finite index is infinite cyclic.)

Let $F$ be the nonabelian free group on two generators $x, y$. For $a, b \in \mathbb{C}$, $|a| = |b| = 1$, let $\vartheta_{a,b}$ be the automorphism of $\mathbb{C}F$ defined by $\vartheta_{a,b}(x) = ax$, $\vartheta_{a,b}(y) = by$. Given $0 \neq \alpha \in \mathbb{C}F$, can we always find $a, b \in \mathbb{C} \setminus \{1\}$ with $|a| = |b| = 1$ such that $\alpha\mathbb{C}F \cap \vartheta_{a,b}(\alpha)\mathbb{C}F \neq 0$?

Let $k$ be a field, let $p$ be a prime, and let $G$ be the Wreath product $\mathbb{Z}_p \wr \mathbb{Z}$ (so the base group has exponent $p$). Does $kG$ have a classical quotient ring? (i.e. do the non-zero-divisors of $kG$ form an Ore set?)

Let $G$ be a sharply doubly transitive permutation group on a set $\Omega$ (see 11.52 for a definition).
$\qquad$ (a) Does $G$ possess a regular normal subgroup if a point stabilizer is locally finite?
$\qquad$ (b) Does $G$ possess a regular normal subgroup if a point stabilizer has an abelian subgroup of finite index?

Construct all non-split extensions of elementary abelian 2-groups $V$ by $H = \text{PSL}_2(q)$ for which $H$ acts irreducibly on $V$.

(Well-known problem). Find an algorithm which decides, by a given finite set of matrices in $SL_3(\mathbb{Z})$, whether the first matrix of this set is contained in the subgroup generated by the remaining matrices. An analogous problem for $SL_4(\mathbb{Z})$ is insoluble since a direct product of two free groups of rank 2 embeds into $SL_4(\mathbb{Z})$.

Does the free group $F_\eta$ ($1 < \eta < \infty$) have a finite subset $S$ for which there is a unique total order of $F_\eta$ making all elements of $S$ positive? Equivalently, does the free representable $l$-group of rank $\eta$ have a basic element? (The analogs for right orders of $F_\eta$ and free $l$-groups have negative answers.)

Is the free $l$-group $\mathcal{F}_\eta$ of rank $\eta$ ($1 < \eta < \infty$) Hopfian? That is, are $l$-homomorphisms from $\mathcal{F}_\eta$ onto itself necessarily one-to-one?

Is it decidable whether or not two elements of a free $l$-group are conjugate?

Is there a normal valued $l$-group $G$ for which there is no abelian $l$-group $A$ with $C(A) \cong C(G)$? (Here $C(G)$ denotes the lattice of convex $l$-subgroups of $G$. If $G$ is not required to be normal valued, this question has an affirmative answer.)

Let $f(n, p)$ be the number of groups of order $p^n$. Is $f(n, p)$ an increasing function of $p$ for any fixed $n \geqslant 5$?

Let $\langle X \mid R \rangle$ be a finite presentation (the words in $R$ are assumed cyclically reduced). Define the length of the presentation to be the sum of the number of generators and the lengths of the relators. Let $F(n)$ be the number of (isomorphism types of) groups that have a presentation of length at most $n$. What can one say about the function $F(n)$? It can be shown that $F(n)$ is not recursive (D. Segal) and it is at least exponential (L. Pyber).

Suppose that a cyclic group $A$ of order 4 is a $TI$-subgroup of a finite group $G$.
$\qquad$ a) Does $A$ localize a component $L$ of $G$ if $A$ intersects $L \cdot C(L)$ trivially?
$\qquad$ b) What is the structure of the normal closure of $A$ in the case where $A$ centralizes each component of $G$?

A generalized quadrangle $GQ(s, t)$ with parameters $s, t$ is by definition an incidence system consisting of points and lines in which every line consists of $s + 1$ points, every two different lines have at most one common point, each point belongs to $t + 1$ lines, and for any point $a$ not lying on a line $L$ there is a unique line containing $a$ and intersecting $L$.
$\qquad$ a) Does $GQ(4, 11)$ exist?
$\qquad$ b) Can the automorphism group of a hypothetical $GQ(5, 7)$ contain involutions?

Does there exist a strongly regular graph with parameters $(85, 14, 3, 2)$ and a non-connected neighborhood of some vertex?

Describe the strongly regular graphs in which the neighborhoods of vertices are generalized quadrangles (see 12.58).

A Frobenius group is a transitive permutation group in which the stabilizer of any two points is trivial. Does there exist a Frobenius group of infinite degree which is primitive as permutation group, and in which the stabilizer of a point is cyclic and has only finitely many orbits?

Conjecture: No. Note that such a group cannot be 2-transitive (Gy. Károlyi, S. J. Kovács, P. P. Pálfy, Aequationes Math., 39 (1990), 161–166; V. D. Mazurov, Siberian Math. J., 31 (1990), 615–617; P. M. Neumann, P. J. Rowley, in: Geometry and cohomology in group theory (London Math. Soc. Lect. Note Ser., 252), Cambridge Univ. Press, 1998, 291–295).

Does there exist a soluble permutation group of infinite degree that has only finitely many orbits on triples? Conjecture: No.

Is it true that for a given number $k \geqslant 2$ and for any (prime) number $n$, there exists a number $N = N(k, n)$ such that every finite group with generators $A = \{a_1, \dots, a_k\}$ has exponent $\leqslant n$ if $(x_1 \dots x_N)^n = 1$ for any $x_1, \dots, x_N \in A \cup \{1\}$?

For given $k$ and $n$, a negative answer implies, for example, the infiniteness of the free Burnside group $B(k, n)$, and a positive answer, in the case of sufficiently large $n$, gives, for example, an opportunity to find a hyperbolic group which is not residually finite (and in this group a hyperbolic subgroup of finite index which has no proper subgroups of finite index).

Let $\mathscr{P} = (P_0, P_1, P_2)$ be a parabolic system in a finite group $G$, belonging to the $C_3$ Coxeter diagram $\underset{1}{\circ}\ \text{–––––} \underset{2}{\circ} \xlongequal{\quad\quad} \underset{3}{\circ}$ and let the Borel subgroup have index at least 3 in $P_0$ and $P_1$. It is known that, if we furthermore assume that the chamber system of $\mathscr{P}$ is geometric and that the projective planes arising as $\{0, 1\}$-residues from $\mathscr{P}$ are desarguesian, then either $G^{(\infty)}$ is a Chevalley group of type $C_3$ or $B_3$ or $G = A_7$. Can we obtain the same conclusion in general, without assuming the previous two hypothesis?

Describe the finite translation planes whose collineation groups act doubly transitively on the set of points of the line at infinity.

Describe the structure of the locally compact soluble groups with the maximum (minimum) condition for closed non-compact subgroups.

Describe the locally compact abelian groups in which any two closed subgroups of finite rank generate a subgroup of finite rank.

Let $F$ be a countably infinite field and $G$ a finite group of automorphisms of $F$. The group ring $\mathbb{Z}G$ acts on $F$ in a natural way. Suppose that $S$ is a subfield of $F$ satisfying the following property: for any $x \in F$ there is a non-zero element $f \in \mathbb{Z}G$ such that $x^f \in S$. Is it true that either $F = S$ or the field extension $F/S$ is purely inseparable? One can show that for an uncountable field an analogous question has an affirmative answer.

Let $p$ be a prime number, $F$ a free pro-$p$-group of finite rank, and $\Delta \neq 1$ an automorphism of $F$ whose order is a power of $p$. Is the rank of $\text{Fix}_F(\Delta) = \{x \in F \mid \Delta(x) = x\}$ finite? If the order of $\Delta$ is prime to $p$, then $\text{Fix}_F(\Delta)$ has infinite rank (W. Herfort, L. Ribes, Proc. Amer. Math. Soc., 108 (1990), 287–295).

Let $d(G)$ denote the smallest cardinality of a generating set of the group $G$. Let $A$ and $B$ be finite groups. Is there a finite group $G$ such that $A, B \leqslant G$, $G = \langle A, B \,\rangle$, and $d(G) = d(A)+d(B)$? The corresponding question has negative answer in the class of solvable groups (L. G. Kovács, H.-S. Sim, Indag. Math., 2, no. 2 (1991), 229–232).

Let $\mathfrak{F}$ be a soluble local hereditary formation of finite groups. Prove that $\mathfrak{F}$ is radical if every finite soluble minimal non-$\mathfrak{F}$-group $G$ is a minimal non-$\mathfrak{N}^{l(G)-1}$-group. Here $\mathfrak{N}$ is the formation of all finite nilpotent groups, and $l(G)$ is the nilpotent length of $G$.

A formation $\mathfrak{H}$ of finite groups is said to have length t if there is a chain of formations $\varnothing = \mathfrak{H}_0 \subset \mathfrak{H}_1 \subset \dots \subset \mathfrak{H}_t = \mathfrak{H}$ in which $\mathfrak{H}_{i-1}$ is a maximal subformation of $\mathfrak{H}_i$. Is the lattice of soluble formations of length $\leqslant 4$ distributive?

Let $\mathfrak{F}$ be a non-primary one-generator composition formation of finite groups. Is it true that if $\mathfrak{F} = \mathfrak{M}\mathfrak{H}$ and the formations $\mathfrak{M}$ and $\mathfrak{H}$ are non-trivial, then $\mathfrak{M}$ is a composition formation?

(B. Jonsson). Is the class $\mathcal{N}$ of the lattices of normal subgroups of groups a variety?

Is every group generated by a set of 3-transpositions locally finite? A set of 3-transpositions is, by definition, a normal set of involutions such that the orders of their pairwise products are at most 3.

(Well-known problem). Does the order (if it is greater than $p^2$) of a finite non-cyclic $p$-group divide the order of its automorphism group?

(M. J. Curran).
a) Does there exist a group of order $p^6$ (here $p$ is a prime number), whose automorphism group has also order $p^6$?
b) Is it true that for $p \equiv 1 \pmod 3$, the smallest order of a $p$-group that is the automorphism group of a $p$-group is $p^7$?
c) The same question for $p = 3$ with $3^9$ replacing $p^7$.

Suppose that $a$ and $b$ are two elements of a finite group $G$ such that the function $$\phi(g) = 1_G(g) - 1^G_{\langle a \rangle}(g) - 1^G_{\langle b \rangle}(g) - 1^G_{\langle ab \rangle}(g) + 2$$ is a character of $G$. Is it true that $G = \langle a, b \rangle$? The converse statement is true.

(K. W. Roggenkamp).
a) Is it true that the number of $p$-blocks of defect 0 of a finite group $G$ is equal to the number of the conjugacy classes of elements $g \in G$ such that the number of solutions of the equation $[x, y] = g$ in $G$ is not divisible by $p$?
b) The same question in the case of $G$ being a simple group.

What is the cardinality of the set of subvarieties of the group variety $\mathfrak{A}_p^3$ (where $\mathfrak{A}_p$ is the variety of abelian groups of prime exponent $p$)?

Find all pairs $(n, r)$ such that the symmetric group $S_n$ contains a maximal subgroup isomorphic to $S_r$.

(Well-known problem). Is it true that if there exist two non-isomorphic groups with the given set of orders of the elements, then there are infinitely many groups with this set of orders of the elements?

Does every variety which is generated by a (known) finite simple group containing a soluble subgroup of derived length $d$ contain a 2-generator soluble group of derived length $d$?

For each known finite simple group, find its maximal 2-generator direct power.

Let $\Gamma$ be a connected undirected graph without loops or multiple edges and suppose that the automorphism group $\operatorname{Aut}(\Gamma)$ acts transitively on the vertex set of $\Gamma$. Is it true that at least one of the following assertions holds?
$\qquad$ 1. The stabilizer of a vertex of $\Gamma$ in $\operatorname{Aut}(\Gamma)$ is finite.
$\qquad$ 2. The group $\operatorname{Aut}(\Gamma)$ as a permutation group on the vertex set of $\Gamma$ admits an imprimitivity system $\sigma$ with finite blocks for which the stabilizer of a vertex of the factor-graph $\Gamma/\sigma$ in $\operatorname{Aut}(\Gamma/\sigma)$ is finite.
$\qquad$ 3. There exists a natural number $n$ such that the graph obtained from $\Gamma$ by adding edges joining distinct vertices the distance between which in $\Gamma$ is at most $n$ contains a regular tree of valency 3.

An undirected graph is called a locally finite Cayley graph of a group $G$ if its vertex set can be identified with the set of elements of $G$ in such a way that, for some finite generating set $X = X^{-1}$ of $G$ not containing 1, two vertices $g$ and $h$ are adjacent if and only if $g^{-1}h \in X$. Do there exist two finitely generated groups with the same locally finite Cayley graph, one of which is periodic and the other is not periodic?

Describe the infinite connected graphs to which the sequences of finite connected graphs with primitive automorphism groups converge. For definitions, see (V. I. Trofimov, Algebra and Logic, 28, no. 3 (1989), 220–237).

Let $G$ be a finitely generated soluble minimax group and let $H$ be a finitely generated residually finite group which has precisely the same finite images as $G$. Must $H$ be a minimax group?

Every metabelian group belonging to a Fitting class of (finite) supersoluble groups is nilpotent. Does the following generalization also hold: Every group belonging to a Fitting class of supersoluble groups has only central minimal normal subgroups?

For a field $K$ of characteristic 2, each finite group $G = \{g_1, \dots, g_n\}$ of odd order $n$ is determined up to isomorphism by its group determinant (Formanek–Sibley, 1990) and even by its reduced norm which is defined as the last coefficient $s_m(x)$ of the minimal polynomial $\phi(\lambda; x) = \lambda^m - s_1(x)\lambda^{m-1} + \dots + (-1)^m s_m(x)$ for the generic element $x = x_1 g_1 + \dots + x_n g_n$ of the group ring $KG$ (Hoehnke, 1991). Is it possible in this theorem to replace $s_m(x)$ by some other coefficients $s_i(x)$, $i < m$? For notation see (G. Frobenius, Sitzungsber. Preuss. Akad. Wiss. Berlin, 1896, 1343–1382) and (K. W. Johnson, Math. Proc. Cambridge Phil. Soc., 109 (1991), 299–311).

Let $N \rightarrowtail G \twoheadrightarrow Q$ be an extension of nilpotent groups, with $Q$ finitely generated, which splits at every prime. Does the extension split? This is known to be true if $N$ is finite or commutative.

Let $G$ be a finitely generated pro-$p$-group not involving the wreath product $C_p \wr \mathbb{Z}_p$ as a closed section (where $C_p$ is a cyclic group of order $p$ and $\mathbb{Z}_p$ is the group of $p$-adic integers). Does it follow that $G$ is $p$-adic analytic?

Let $G$ be a finitely generated pro-$p$-group and let $g_1, \dots, g_n \in G$. Let $H$ be an open subgroup of $G$, and suppose there exists a non-trivial word $w = w(X_1, \dots, X_n)$ such that $w(a_1, \dots, a_n) = 1$ whenever $a_1 \in g_1 H, \dots, a_n \in g_n H$ (that is, $G$ satisfies a coset identity). Does it follow that $G$ satisfies some non-trivial identity?

A positive answer to this question would imply that an analogue of the Tits Alternative holds for finitely generated pro-$p$-groups. Note that J. S. Wilson and E. I. Zelmanov (J. Pure Appl. Algebra, 81 (1992), 103–109) showed that the graded Lie algebra $L_p(G)$ with respect to the dimension subgroups of $G$ in characteristic $p$ satisfies a polynomial identity.

Find a non-empty Fitting class $\mathfrak{F}$ and a non-soluble finite group $G$ such that $G$ has no $\mathfrak{F}$-injectors.

Let $\mathfrak{F}$ be the formation of all finite groups all of whose composition factors are isomorphic to some fixed simple non-abelian group $T$. Prove that $\mathfrak{F}$ is indecomposable into a product of two non-trivial subformations.

Let $F$ be a free group of finite rank, $R$ its recursively defined normal subgroup. Is it true that
$\qquad$ a) the word problem for $F/R$ is soluble if and only if it is soluble for $F/[R, R]$?
$\qquad$ b) the conjugacy problem for $F/R$ is soluble if and only if it is soluble for $F/[R, R]$?
$\qquad$ c) the conjugacy problem for $F/[R, R]$ is soluble if the word problem is soluble for $F/[R, R]$?

Is every periodic group with a regular automorphism of order 4 locally finite?

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.

Does there exist a simple $T_0$-group?

Is every proper factor-group of a group of Golod (see 9.76) residually finite?