Let $p$ be a prime number. Describe the groups of order $p^9$ of nilpotency class 2 which contain subgroups $X$ and $Y$ such that $|X| = |Y| = p^3$ and any non-identity elements $x \in X$, $y \in Y$ do not commute. An answer to this question would yield a description of the semifields of order $p^3$.

A mixed identity of a group $G$ is, by definition, an identity of an algebraic system obtained from $G$ by supplementing its signature by some set of 0-ary operations. One can develop a theory of mixed varieties of groups on the basis of this notion (see, for example, V. S. Anashin, Math. USSR Sbornik, 57 (1987), 171–182).

Construct an example of a class of groups which is not a mixed variety, but which is closed under taking factor-groups, Cartesian products, and those subgroups of Cartesian powers that contain the diagonal subgroup.

Characterize (in terms of bases of mixed identities (see 10.2) or in terms of generating groups) minimal mixed varieties of groups.

Let $p$ be a prime number. Is it true that a mixed variety of groups (see 10.2) generated by an arbitrary finite $p$-group of sufficiently large nilpotency class is a variety of groups?

Construct an example of a finite group whose mixed identities (see 10.2) do not have a finite basis. Is the group constructed in (R. Bryant, Bull. London Math. Soc., 14, no. 2 (1982), 119–123) such an example?

Is it true that in an abelian group every non-discrete group topology can be strengthened up to a non-discrete group topology such that the group becomes a complete topological group?

Is it true that in a countable group $G$, any non-discrete group topology satisfying the first axiom of countability can be strengthened up to a non-discrete group topology such that $G$ becomes a complete topological group?

Does there exist a topological group which cannot be embedded in the multiplicative semigroup of a topological ring?

Let $p$ be a prime number and let $L_p$ denote the set of all quasivarieties each of which is generated by a finite $p$-group. Is $L_p$ a sublattice of the lattice of all quasivarieties of groups?

Is it true that a quasivariety generated by a finitely generated torsion-free soluble group and containing a non-abelian free metabelian group, can be defined in the class of torsion-free groups by an independent system of quasiidentities?

Is it true that every finitely presented group contains either a free subsemigroup on two generators or a nilpotent subgroup of finite index?

Does there exist a finitely generated semigroup $S$ with cancellation having non-exponential growth and such that its group of left quotients $G = S^{-1}S$ (which exists) is a group of exponential growth? An affirmative answer would give a positive solution to Problem 12 in (S. Wagon, The Banach–Tarski Paradox, Cambridge Univ. Press, 1985).

Does there exist a massive set of independent elements in a free group $F_2$ on free generators $a, b$, that is, a set $E$ of irreducible words on the alphabet $a, b, a^{-1}, b^{-1}$ such that
$\qquad$ 1) no element $w \in E$ belongs to the normal closure of $E \setminus \{w\}$, and
$\qquad$ 2) the massiveness condition is satisfied: $\overline{\lim}\limits_{n \to \infty} \sqrt[n]{\lvert E_n \rvert} = 3$, where $E_n$ is the set of all words of length $n$ in $E$?

a) Does every group satisfying the minimum condition on subgroups satisfy the weak maximum condition on subgroups?
b) Does every group satisfying the maximum condition on subgroups satisfy the weak minimum condition on subgroups?

For every (known) finite quasisimple group and every prime $p$, find the faithful $p$-modular absolutely irreducible linear representations of minimal degree.

A class of groups is called a direct variety if it is closed under taking subgroups, factor-groups, and direct products (Yu. M. Gorchakov, Groups with Finite Classes of Conjugate Elements, Moscow, Nauka, 1978 (Russian)). It is obvious that the class of $FC$-groups is a direct variety. P. Hall (J. London Math. Soc., 34, no. 3 (1959), 289–304) showed that the class of finite groups and the class of abelian groups taken together do not generate the class of $FC$-groups as a direct variety, and it was shown in (L. A. Kurdachenko, Ukrain. Math. J., 39, no. 3 (1987), 255–259) that the direct variety of $FC$-groups is also not generated by the class of groups with finite derived subgroups. Is the direct variety of $FC$-groups generated by the class of groups with finite derived subgroups together with the class of $FC$-groups having quasicyclic derived subgroups?

(M. J. Tomkinson). Let $G$ be an $FC$-group whose derived subgroup is embeddable in a direct product of finite groups. Must $G/Z(G)$ be embeddable in a direct product of finite groups?

(M. J. Tomkinson). Let $G$ be an $FC$-group which is residually in the class of groups with finite derived subgroups. Must $G/Z(G)$ be embeddable in a direct product of finite groups?

Characterize the radical associative rings such that the set of all normal subgroups of the adjoint group coincides with the set of all ideals of the associated Lie ring.

In a Chevalley group of rank $\leqslant 6$ over a finite field of order $\leqslant 9$, describe all subgroups of the form $H = \langle H \cap U, H \cap V \rangle$ that are not contained in any proper parabolic subgroup, where $U$ and $V$ are opposite unipotent subgroups.

Is it true that extraction of roots in braid groups is unique up to conjugation?

A braid is said to be coloured if its strings represent the identity permutation. Is it true that links obtained by closing coloured braids are equivalent if and only if the original braids are conjugate in the braid group?

(Well-known problem). Does there exist an algorithm which decides for a given automorphism of a free group whether this automorphism has a non-trivial fixed point?

a) Does there exist an algorithm which decides, for given elements $a, b$ and an automorphism $\phi$ of a free group, whether the equation $a x^\phi = xb$ is soluble in this group? This question seems to be useful for solving the problem of equivalence of two knots.
b) Does there exist an algorithm which decides whether the equation of the form $w(x_{i_1}^{\varphi_1}, \dots, x_{i_n}^{\varphi_n}) = 1$ is soluble in a free group where $\varphi_1, \dots, \varphi_n$ are automorphisms of this group?

a) Let $t$ be an involution of a finite group $G$ and suppose that the set $D = t^G \cup \{t^x t^y \mid x, y \in G, \ \lvert t^x t^y \rvert = 2\}$ does not intersect $O_2(G)$. Prove that if $t \in O_2(C(d))$ for any involution $d$ from $C_D(t)$, then $D = t^G$ (and in this case the structure of the group $\langle D \rangle$ is known).
b) A significantly more general question. Let $t$ be an involution of a finite group $G$ and suppose that $t \in Z^*(N(X))$ for every non-trivial subgroup $X$ of odd order which is normalized, but not centralized, by $t$. What is the group $\langle t^G \rangle$?

Is it true that finite strongly regular graphs with $\lambda = 1$ have rank 3?

Describe the finite groups which contain a set of involutions $D$ such that, for any subset $D_0$ of $D$ generating a 2-subgroup, the normalizer $N_D(D_0)$ also generates a 2-subgroup.

Does there exist a non-right-orderable group which is residually finite $p$-group for a finite set of prime numbers $p$ containing at least two different primes? If a group is residually finite $p$-group for an infinite set of primes $p$, then it admits a linear order (A. H. Rhemtulla, Proc. Amer. Math. Soc., 41, no. 1 (1973), 31–33).

Suppose that $G$ is an algebraic group, $H$ is a closed normal subgroup of $G$ and $f : G \to G/H$ is the canonical homomorphism. What conditions ensure that there exists a rational section $s : G/H \to G$ such that $sf = 1$?

A group word is said to be universal on a group G if its values on $G$ run over the whole of $G$. For an arbitrary constant $c > 8/5$, J. L. Brenner, R. J. Evans, and D. M. Silberger (Proc. Amer. Math. Soc., 96, no. 1 (1986), 23–28) proved that there exists a number $N_0 = N_0(c)$ such that the word $x^r y^s$, $rs \neq 0$, is universal on the alternating group $\mathbb{A}_n$ for any $n \geqslant \max \{N_0, \ c \cdot \log m(r, s)\}$, where $m(r, s)$ is the product of all primes dividing $rs$ if $r, s \notin \{-1, 1\}$, and $m(r, s) = 1$ if $r, s \in \{-1, 1\}$. For example, one can take $N_0(5/2) = 5$ and $N_0(2) = 29$. Find an analogous bound for the degree of the symmetric group $S_n$ under hypothesis that at least one of the $r, s$ is odd: up to now, here only a more crude bound $n \geqslant \max \{6, \ 4m(r, s) - 4\}$ is known (M. Droste, Proc. Amer. Math. Soc., 96, no. 1 (1986), 18–22).

For HNN-extensions of the form $G = \langle t, A \mid t^{-1}Bt = C, \phi \rangle$, where $A$ is a finitely generated abelian group and $\phi : B \to C$ is an isomorphism of two of its subgroups, find
$\qquad$ a) a criterion to be residually finite;
$\qquad$ b) a criterion to be Hopfian.

Does there exist a non-soluble finite group which coincides with the product of any two of its non-conjugate maximal subgroups?

Is it true that every finitely generated torsion-free subgroup of $GL_n(\mathbb{C})$ is residually in the class of torsion-free subgroups of $GL_n(\mathbb{Q})$?

Is it true that $SL_n(\mathbb{Z}[x_1, \dots, x_r])$, for $n$ sufficiently large, is a group of type $(FP)_m$ (which is defined in the same way as $(FP)_\infty$ was defined in 6.3, but with that weakening that the condition of being finitely generated is not imposed on the terms of the resolution with numbers $> m$)?

Suppose that $G$ is a finitely generated metabelian group all of whose integral homology groups are finitely generated. Is it true that $G$ is a group of finite rank? The answer is affirmative if $G$ splits over the derived subgroup (J. R. J. Groves, Quart. J. Math., 33, no. 132 (1982), 405–420).

(W. van der Kallen). Does $E_{n+3}(\mathbb{Z}[x_1, \dots, x_n])$, $n \geqslant 1$, have finite breadth with respect to the set of transvections?

a) Is the membership problem soluble for the subgroup $E_n(R)$ of $SL_n(R)$ where $R$ is a commutative ring?
b) Is the word problem soluble for the groups $K_i(R)$ where $K_i$ are the Quillen $K$-functors and $R$ is a commutative ring?

(H. Bass). Let $G$ be a group, $e$ an idempotent matrix over $\mathbb{Z}G$ and let $\operatorname{tr} e = \sum_{g \in G} e_g g$, $e_g \in \mathbb{Z}$.
$\qquad$ Strong conjecture: for any non-trivial $x \in G$, the equation $\sum_{g \sim x} e_g = 0$ holds where $\sim$ denotes conjugacy in $G$.
$\qquad$ Weak conjecture: $\sum_{g \in G} e_g = 0$.

(Well-known problem). Let $\Gamma$ be an almost polycyclic group with no non-trivial finite normal subgroups, and let $k$ be a field. The complete ring of quotients $Q(k\Gamma)$ is a matrix ring $M_n(D)$ over a skew field. Conjecture: $n$ is the least common multiple of the orders of the finite subgroups of $\Gamma$. An equivalent formulation (M. Lorenz) is as follows: $\rho(G_0(k\Gamma)) = \rho(G_0(k\Gamma)_{\mathcal{F}})$, where $G_0(k\Gamma)$ is the Grothendieck group of the category of finitely-generated $k\Gamma$-modules, $G_0(k\Gamma)_{\mathcal{F}}$ is the subgroup generated by classes of modules induced from finite subgroups of $\Gamma$, and $\rho$ is the Goldie rank. There is a stronger conjecture: $G_0(k\Gamma) = G_0(k\Gamma)_{\mathcal{F}}$.

(J. T. Stafford). Let $G$ be a poly-$\mathbb{Z}$-group and let $k$ be a field. Is it true that every finitely generated projective $kG$-module is either a free module or an ideal?

Let $R$ and $S$ be associative rings with identity such that 2 is invertible in $S$. Let $\Lambda_I : GL_n(R) \to GL_n(R/I)$ be the homomorphism corresponding to an ideal $I$ of $R$ and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by elementary transvections $t_{ij}(x)$. Let $a_{ij} = t_{ij}(1)t_{ji}(-1)t_{ij}(1)$, let the bar denote images in the factor-group of $GL_n(R)$ by the centre and let $PG = \overline{G}$ for $G \leqslant GL_n(R)$. A homomorphism
$$\Lambda : E_n(R) \to GL(W) = GL_m(S)$$ is called standard if $S^m = P \oplus \dots \oplus P \oplus Q$ (a direct sum of $S$-modules in which there are $n$ summands $P$) and
$$\Lambda x = g^{-1}\tau(\delta^*(x)f + ({}^t\delta^*(x)^\nu)^{-1}(1 - f))g, \quad x \in E_n(R),$$ where $\delta^* : GL_n(R) \to GL_n(\operatorname{End} P)$ is the homomorphism induced by a ring homomorphism $\delta : R \to \operatorname{End} P$ taking identity to identity, $g$ is an isomorphism of the module $W$ onto $S^m$, $\tau : GL_n(\operatorname{End} P) \to GL(gW)$ is an embedding, $f$ is a central idempotent of $\delta R$, $t$ denotes transposition and $\nu$ is an antiisomorphism of $\delta R$. Let $n \geqslant 3$, $m \geqslant 2$. One can show that the homomorphism
$$\Lambda_0 : PE_n(R) \to GL(W) = GL_m(S)$$ is induced by a standard homomorphism $\Lambda$ if $\Lambda_0 \bar{a}_{ij} = g^{-1}\tau(a_{ij}^*)g$ for some $g$ and $\tau$ and for any $i \neq j$ where $a_{ij}^*$ denotes the matrix obtained from $a_{ij}$ by replacing 0 and 1 from $R$ by 0 and 1 from $\operatorname{End} P$. Find (at least in particular cases) the form of a homomorphism $\Lambda_0$ that does not satisfy the last condition.

Prove that every standard homomorphism (see 10.43 for definition) $\Lambda$ of $E_n(R) \subset GL_n(R) = GL(V)$ into $E_m(S) \subset GL_m(S) = GL(W)$ originates from a collineation and a correlation, that is, has the form $\Lambda x = (g^{-1}xg)f + (h^{-1}xh)(1 - f)$, where $g$ is a semilinear isomorphism $V_R \to W_S$ (collineation) and $h$ is a semilinear isomorphism $V_R \to {}_S W'_{S_0}$ (correlation).

Let $n \geqslant 3$ and suppose that $N$ is a subgroup of $GL_n(R)$ which is normalized by $E_n(R)$. Prove that either $N$ contains $E_n(R)$ or $\Lambda_I[N, E_n(R)] = 1$ for a suitable ideal $I \neq R$ of $R$.

Prove that for every element $\sigma \in GL_n(R)$, $n \geqslant 3$, there exist transvections $\tau_1, \tau_2$ such that $[[\sigma, \tau_1], \tau_2]$ is a unipotent element.

Describe the automorphisms of $PE_2(R)$ in the case where the ring $R$ is commutative and 2 and 3 have inverses in it.

Let $V$ be a vector space of finite dimension over a field of prime order. A subset $R$ of $\text{GL}(V) \cup \{0\}$ is called regular if $|R| = |V|$, $0, 1 \in R$ and $vx \neq vy$ for any non-trivial vector $v \in V$ and any distinct elements $x, y \in R$. It is obvious that $\tau, \varepsilon, \mu_g$ transform a regular set into a regular one, where $x^\tau = x^{-1}$ for $x \neq 0$ and $0^\tau = 0$, $x^\varepsilon = 1 - x$, $x^{\mu_g} = xg^{-1}$ and $g$ is a non-zero element of the set being transformed. We say that two regular subsets are equivalent if one can be obtained from the other by a sequence of such transformations.
$\qquad$ a) Study the equivalence classes of regular subsets.
$\qquad$ b) Is every regular subset equivalent to a subgroup of $\text{GL}(V)$ together with 0?

Does there exist a group $G$ satisfying the following four conditions:
$\qquad$ 1) $G$ is simple, moreover, there is an integer $n$ such that $G = C^n$ for any conjugacy class $C$,
$\qquad$ 2) all maximal abelian subgroups of $G$ are conjugate in $G$,
$\qquad$ 3) every maximal abelian subgroup of $G$ is self-normalizing and it is the centralizer of any of its nontrivial elements,
$\qquad$ 4) there is an integer $m$ such that if $H$ is a maximal abelian subgroup of $G$ and $a \in G \setminus H$ then every element in $G$ is a product of $m$ elements in $aH$?

Complex characters of a finite group $G$ induced by linear characters of cyclic subgroups are called induced cyclic characters of $G$. Computer-aided computations (A. V. Rukolaine, Abstracts of the 10th All–Union Sympos. on Group Theory, Gomel’, 1986, p. 199 (Russian)) show that there exist groups (for example, $\mathbb{S}_5$, $SL_2(13)$, $M_{11}$) all of whose irreducible complex characters are integral linear combinations of induced cyclic characters and the principal character of the group. Describe all finite groups with this property.

Describe the structure of thin abelian $p$-groups. Such a description is known in the class of separable $p$-groups (C. Megibben, Mich. Math. J., 13, no. 2 (1966), 153–160).

(R. Mines). It is well known that the topology on the completion of an abelian group under the $p$-adic topology is the $p$-adic topology. R. Warfield has shown that this is also true in the category of nilpotent groups. Find a categorical setting for this theorem which includes the case of nilpotent groups.

(M. Dugas). Let $\mathfrak{R}$ be the Reid class, i. e. the smallest containing $\mathbb{Z}$ and closed under direct sums and direct products. Is $\mathfrak{R}$ closed under direct summands?

(R. Göbel). For a cardinal number $\mu$, let
$$\mathbb{Z}^{<\mu} = \{f \in \mathbb{Z}^\mu \mid \lvert\operatorname{supp} (f)\rvert < \mu\} \quad \text{and} \quad G_\mu = \mathbb{Z}^\mu / \mathbb{Z}^{<\mu}.$$ $\qquad$ a) Find a non-zero direct summand $D$ of $G_{\omega_1}$ such that $D \ncong G_{\omega_1}$.
$\qquad$ b) Investigate the structure of $G_\mu$ (the structure of $G_{\omega_0}$ is well known).

(A. Mader). “Standard B”, that is, $B = \mathbb{Z}(p) \oplus \mathbb{Z}(p^2) \oplus \dots$, is slender as a module over its endomorphism ring (A. Mader, in: Abelian Groups and Modules, Proc., Udine, 1984, Springer, 1984, 315–327). Which abelian $p$-groups are slender as modules over their endomorphism rings?

Is the lattice of formations of finite nilpotent groups of class $\leqslant 4$ distributive?

What are the minimal non-$\mathfrak{A}$-formations? An $\mathfrak{A}$-formation is, by definition, the formation of the finite groups all of whose Sylow subgroups are abelian.

Is the subsemigroup generated by the undecomposable formations in the semigroup of formations of finite groups free?

Is a $p'$-group $G$ locally nilpotent if it admits a splitting automorphism $\varphi$ of prime order $p$ such that all subgroups of the form $\langle g, g^\varphi, \dots, g^{\varphi^{p-1}} \rangle$ are nilpotent? An automorphism $\varphi$ of order $p$ is called splitting if $g g^\varphi g^{\varphi^2} \dots g^{\varphi^{p-1}} = 1$ for all $g \in G$.

Does every periodic group ($p$-group) $A$ of regular automorphisms of an abelian group have non-trivial centre?

Suppose that $H$ is a proper subgroup of a group $G$, $a \in H$, $a^2 \neq 1$ and for every $g \in G \setminus H$ the subgroup $\langle a, a^g \rangle$ is a Frobenius group whose complement contains $a$. Does the set-theoretic union of the kernels of all Frobenius subgroups of $G$ with complement $\langle a \rangle$ constitute a subgroup? For definitions see 6.55; see also (A. I. Sozutov, Algebra and Logic, 34, no. 5 (1995), 295–305).

Construct an example of a (periodic) group without subgroups of index 2 which is generated by a conjugacy class of involutions $X$ such that the order of the product of any two involutions from $X$ is odd.

Is there a doubly transitive permutation group in which the stabilizer of a point is infinite cyclic?

Does there exist a non-periodic doubly transitive permutation group with a periodic stabilizer of a point?

Determine the structure of infinite 2-transitive permutation groups $(G, \Omega)$ in which the stabilizer of a point $\alpha \in \Omega$ has the form $G_\alpha = A \cdot G_{\alpha\beta}$ where $G_{\alpha\beta}$ is the stabilizer of two points $\alpha, \beta, \alpha \neq \beta$, such that $G_{\alpha\beta}$ contains an element inverting the subgroup $A$. Suppose, in particular, that $A \setminus \{1\}$ contains at most two conjugacy classes of $G_\alpha$; does $G$ then possess a normal subgroup isomorphic to $PSL_2$ over a field?

Is a group $G$ non-simple if it contains two non-trivial subgroups $A$ and $B$ such that $AB \neq G$ and $AB^g = B^gA$ for any $g \in G$? This is true if $G$ is finite (O. H. Kegel, Arch. Math., 12, no. 2 (1961), 90–93).

The class $LN\mathfrak{M}_p$ of locally nilpotent groups admitting a splitting automorphism of prime order $p$ (for definition see 10.59) is a variety of groups with operators (E. I. Khukhro, Math. USSR Sbornik, 58 (1987), 119–126). Is it true that
$$LN\mathfrak{M}_p = (\mathfrak{N}_{c(p)} \cap LN\mathfrak{M}_p) \vee (\mathfrak{B}_p \cap LN\mathfrak{M}_p)$$ where $\mathfrak{N}_{c(p)}$ is the variety of nilpotent groups of some $p$-bounded class $c(p)$ and $\mathfrak{B}_p$ is the variety of groups of exponent $p$?

Find a geometrical justification for the Whitehead method for free products similar to the substantiation given for free groups by Whitehead himself and more visual than given in (D. J. Collins, H. Zieschang, Math. Z., 185, no. 4 (1984), 487–504; 186, no. 3 (1984), 335–361).

Is it true that the centralizer of any automorphism (any finite set of automorphisms) in the automorphism group of a free group of finite rank is a finitely presented group? This is true in the case of rank 2 and in the case of inner automorphisms for any finite rank.

Enumerate all formations of finite groups all of whose subformations are $S_n$-closed.

Suppose that a group $G$ contains an element $a$ of prime order such that its centralizer $C_G(a)$ is finite and all subgroups $\langle a, a^g \rangle$, $g \in G$, are finite and almost all of them are soluble. Is $G$ locally finite? This problem is closely connected with 6.56.

Suppose that a group $G$ contains an element $a$ of prime order $p$ such that the normalizer of every finite subgroup containing $a$ has finite periodic part and all subgroups $\langle a, a^g \rangle$, $g \in G$, are finite and almost all of them are soluble. Does $G$ possess a periodic part if $p > 2$?

Suppose that $G$ is a periodic group containing an elementary abelian subgroup $R$ of order 4. Must $G$ be locally finite
$\qquad$ a) if $C_G(R)$ is finite?
$\qquad$ b) if the centralizer of every involution of $R$ in $G$ is a Chernikov group?

Does there exist a non-Chernikov group which is a product of two Chernikov subgroups?