(I. M. Isaacs). Does there exist a finite group partitioned by subgroups of equal order not all of which are abelian? (Cf. 15.26.)

(P. Schmid). Does there exist a finite non-abelian $p$-group $G$ such that $H^1(G/\Phi(G), Z(\Phi(G))) = 0$?

Let $G$ be a group in which every 4-element subset contains two elements generating a nilpotent subgroup. Is it true that every 2-generated subgroup of $G$ is nilpotent?

Let $x$ be a right 4-Engel element of a group $G$.
$\qquad$ a) Is it true that the normal closure $\langle x \rangle^G$ of $x$ in $G$ is nilpotent if $G$ is locally nilpotent?
$\qquad$ b) If the answer to a) is affirmative, is there a bound on the nilpotency class of $\langle x \rangle^G$?
$\qquad$ c) Is it true that $\langle x \rangle^G$ is always nilpotent?

Is the nilpotency class of a nilpotent group generated by $d$ left 3-Engel elements bounded in terms of $d$? A group generated by two left 3-Engel elements is nilpotent of class at most 4 (J. Pure Appl. Algebra, 188 (2004), 1–6.)

a) Is there a function $f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ such that every nilpotent group generated by $d$ left $n$-Engel elements is nilpotent of class at most $f(n, d)$?
b) Is there a function $g: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ such that every nilpotent group generated by $d$ right $n$-Engel elements is nilpotent of class at most $g(n, d)$?

a) Is there a group in which the set of right Engel elements does not form a subgroup?
b) Is there a group in which the set of bounded right Engel elements does not form a subgroup?

a) Is there a group containing a right Engel element which is not a left Engel element?
b) Is there a group containing a bounded right Engel element which is not a left Engel element?

Is there a group containing a left Engel element whose inverse is not a left Engel element?

Is there a group containing a right 4-Engel element which does not belong to the Hirsch–Plotkin radical?

Is there a group containing a left 3-Engel element which does not belong to the Hirsch–Plotkin radical?

Are there functions $e, c : \mathbb{N} \to \mathbb{N}$ such that if in a nilpotent group $G$ a normal subgroup $H$ consists of right $n$-Engel elements of $G$, then $H^{e(n)} \leqslant \zeta_{c(n)}(G)$?

Let $G$ be a totally imprimitive $p$-group of finitary permutations on an infinite set. Suppose that the support of any cycle in the cyclic decomposition of every element of $G$ is a block for $G$. Does $G$ necessarily contain a minimal non-$FC$-subgroup?

Can the braid group $B_n$ for $n \geqslant 4$ be embedded into the automorphism group $\text{Aut}(F_{n-2})$ of a free group $F_{n-2}$ of rank $n - 2$?
Artin's theorem implies that $B_n$ can be embedded into $\text{Aut}(F_n)$, and an embedding of $B_n$, $n \geqslant 3$, into $\text{Aut}(F_{n-1})$ was constructed in (B. Perron, J. P. Vannier, Math. Ann., 306 (1996), 231–245).

Construct an algorithm which, for a given polynomial $f \in \mathbb{Z}[x_1, x_2, \dots, x_n]$, finds (explicitly, in terms of generators) the maximal subgroup $G_f$ of the group $\text{Aut}(\mathbb{Z}[x_1, x_2, \dots, x_n])$ that leaves $f$ fixed. This question is related to description of the solution set of the Diophantine equation $f = 0$.

Let $A$ be an Artin group of finite type, that is, the corresponding Coxeter group is finite. It is known that the group $A$ is linear (S. Bigelow, J. Amer. Math. Soc., 14 (2001), 471–486; D. Krammer, Ann. Math., 155 (2002), 131–156; F. Digne, J. Algebra, 268 (2003), 39–57; A. M. Cohen, D. B. Wales, Israel J. Math., 131 (2002), 101–123). Is it true that the automorphism group $\text{Aut}(A)$ is also linear?

If a finitely generated group $G$ has $n < \infty$ maximal subgroups, must $G$ be finite? In particular, what if $n = 3$?

Let $\mathbf{A}$ be the class of compact groups $A$ with the property that whenever two compact groups $B$ and $C$ contain $A$, they can be embedded in a common compact group $D$ by embeddings agreeing on $A$. I showed (Manuscr. Math., 58 (1987) 253–281) that all members of $\mathbf{A}$ are (not necessarily connected) finite-dimensional compact Lie groups satisfying a strong “local simplicity” property, and that all finite groups do belong to $\mathbf{A}$.
$\qquad$ a) Is it true that $\mathbb{R}/\mathbb{Z} \in \mathbf{A}$?
$\qquad$ b) Do any nonabelian connected compact Lie groups belong to $\mathbf{A}$?
$\qquad$ c) If $A$ belongs to $\mathbf{A}$, must the connected component of the identity in $A$ belong to $\mathbf{A}$?

If $F$ is a free group of finite rank, $R$ a retract of $F$, and $H$ a subgroup of $F$ of finite rank, must $H \cap R$ be a retract of $H$?

If $M$ is a real manifold with nonempty boundary, and $G$ the group of self-homeomorphisms of $M$ which fix the boundary pointwise, is $G$ right-orderable?

a) If $A, B, C$ are torsion-free abelian groups with $A \cong A \oplus B \oplus C$, must $A \cong A \oplus B$?
b) What if, furthermore, $B \cong C$?

Suppose $A$ is a group which belongs to a variety $\mathfrak{V}$ of groups, and which is embeddable in the full symmetric group $S$ on an infinite set. Must the coproduct in $\mathfrak{V}$ of two copies of $A$ also be embeddable in $S$? (N. G. de Bruijn proved that this is true if $\mathfrak{V}$ is the variety of all groups.)

Suppose the full symmetric group $S$ on a countably infinite set is generated by the union of two subgroups $G$ and $H$. Must $S$ be finitely generated over one of these subgroups?

(A. Blass, J. Irwin, G. Schlitt). Does $\mathbb{Z}^\omega$ have a subgroup whose dual is free of uncountable rank?

(S. P. Farbman). Let $X$ be the set of complex numbers $\alpha$ such that the group generated by the two $2 \times 2$ matrices $I + \alpha e_{12}$ and $I + e_{21}$ is not free on those generators.
$\qquad$ a) Does $X$ contain all rational numbers in the interval $(-4, 4)$?
$\qquad$ b) Does $X$ contain any rational number in the interval $[27/7, 4)$?

Cf. 4.41, 15.83, and (S. P. Farbman, Publ. Mat., 39 (1995) 379–391).

Are the classes of right-orderable and right-ordered groups closed under taking solutions of equations $w(a_1, \dots, a_k, x_1, \dots, x_n) = 1$? (Here the closures are under group embeddings and order-preserving embeddings, respectively.) This is true for equations with a single constant, when $k = 1$ (J. Group Theory, 11 (2008), 623–633).

Can the free product of two ordered groups with order-isomorphic amalgamated subgroups be lattice orderable but not orderable?

Is there a soluble right-orderable group with insoluble word problem?

Construct a finitely presented orderable group with insoluble word problem.

Is there a constant $l$ such that every finitely presented soluble group has a subgroup of finite index of nilpotent length at most $l$? Cf. 16.35.

Can a soluble right-orderable group have finite quotient by the derived subgroup?

Is the following analogue of the Cayley–Hamilton theorem true for the free group $F_n$ of rank $n$: If $w \in F_n$ and $\varphi \in \text{Aut}\,F_n$ are such that $\langle w, \varphi(w), \dots, \varphi^n(w) \rangle = F_n$, then $\langle w, \varphi(w), \dots, \varphi^{n-1}(w) \rangle = F_n$?

Can the quasivariety generated by the group $\langle a, b \mid a^{-1} ba = b^{-1} \rangle$ be defined by a set of quasi-identities in a finite set of variables? The answer is known for all other groups with one defining relation.

Let $\mathfrak{N}_c$ be the quasivariety of nilpotent torsion-free groups of class at most $c$. Is it true that the dominion in $\mathfrak{N}_c$ (see the definition in 16.20) of a divisible subgroup in every group in $\mathfrak{N}_c$ is equal to this subgroup?

Suppose we have a finite two-colourable triangulation of the sphere, with triangles each coloured either black or white so that no pair of triangles with the same colour share an edge. On each vertex we write a generator, and we assume the generators commute. We use these generators to generate an abelian group $G_W$ with relations stating that the generators around each white triangle add to zero. Doing the same thing with the black triangles, we generate a group $G_B$.

Conjecture: $G_W$ is isomorphic to $G_B$.

Two groups are called isospectral if they have the same set of element orders. Find all finite non-abelian simple groups $G$ for which there is a finite group $H$ isospectral to $G$ and containing a proper normal subgroup isomorphic to $G$. For every simple group $G$ determine all groups $H$ satisfying this condition.

It is easy to show that a group $H$ must satisfy the condition $G < H \leqslant \text{Aut } G$.

Is there an integer $n$ such that for all $m > n$ the alternating group $A_m$ has no non-trivial $A_m$-permutable subgroups? (See the definition in 17.112.)

A formation $\mathfrak{F}$ is called radical in a class $\mathfrak{H}$ if $\mathfrak{F} \subseteq \mathfrak{H}$ and in every $\mathfrak{H}$-group the product of any two normal $\mathfrak{F}$-subgroups belongs to $\mathfrak{F}$. Let $\mathfrak{M}$ be the class of all saturated hereditary formations of finite groups such that the formation of all finite supersoluble groups is radical in every element of $\mathfrak{M}$. Is it true that $\mathfrak{M}$ has the largest (by inclusion) element?

Is there a positive integer $n$ such that the hypercenter of any finite soluble group coincides with the intersection of $n$ system normalizers of that group? What is the least number with this property?

Let $N$ be a nilpotent subgroup of a finite group $G$. Do there always exist elements $x, y \in G$ such that $N \cap N^x \cap N^y \leqslant F(G)$?

Let $S$ be a solvable subgroup of a finite group $G$ that has no nontrivial solvable normal subgroups.
$\qquad$ a) (L. Babai, A. J. Goodman, L. Pyber). Do there always exist seven conjugates of $S$ whose intersection is trivial?
$\qquad$ b) Do there always exist five conjugates of $S$ whose intersection is trivial?

Let $\overline{G}$ be a simple algebraic group of adjoint type over the algebraic closure $\mathbb{F}_p$ of a finite field $\mathbb{F}_p$ of prime order $p$, and $\sigma$ a Frobenius map (that is, a surjective homomorphism such that $G_\sigma = C_{\overline{G}}(\sigma)$ is finite). Then $G = O^{p'}(G_\sigma)$ is a finite group of Lie type. For a maximal $\sigma$-stable torus $T$ of $\overline{G}$, let $N = N_{\overline{G}}(T) \cap G$. Assume also that $G$ is simple and $G \not\cong \text{SL}_3(2)$. Does there always exist $x \in G$ such that $N \cap N^x$ is a $p$-group?

Let $\pi$ be a set of primes. Find all finite simple $D_\pi$-groups (see 3.62) in which
$\qquad$ a) every subgroup is a $D_\pi$-group (H. Wielandt);
$\qquad$ b) every subgroup possessing a Hall $\pi$-subgroup is a $D_\pi$-group.

Let $\pi$ be a set of primes. A finite group is called a $C_\pi$-group if it possesses exactly one class of conjugate Hall $\pi$-subgroups. A finite group is called a $D_\pi$-group if any two of its maximal $\pi$-subgroups are conjugate.
$\qquad$ a) In a $C_\pi$-group, is an overgroup of a Hall $\pi$-subgroup always a $C_\pi$-group? — An affirmative answer in the case $2 \notin \pi$ follows mod CFSG from (F. Gross, Bull. London Math. Soc., 19, no. 4 (1987), 311–319).
$\qquad$ b) In a $D_\pi$-group, is an overgroup of a Hall $\pi$-subgroup always a $D_\pi$-group?

A subgroup $H$ of a group $G$ is called pronormal if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. We say that $H$ is strongly pronormal if $L^g$ is conjugate to a subgroup of $H$ in $\langle H, L^g \rangle$ for every $L \leqslant H$ and $g \in G$.
$\qquad$ a) In a finite simple group, are Hall subgroups always pronormal?
$\qquad$ b) In a finite simple group, are Hall subgroups always strongly pronormal?
$\qquad$ c) In a finite group, is a Hall subgroup with a Sylow tower always strongly pronormal?

Notice that there exist finite (non-simple) groups with a non-pronormal Hall subgroup. Hall subgroups with a Sylow tower are known to be pronormal.

Let $G$ be a finite $p$-group in which every 2-generator subgroup has cyclic derived subgroup. Is the derived length of $G$ bounded?
If $p \neq 2$, then $G^{(2)} = 1$ (J. Alperin), but for $p = 2$ there are examples with $G^{(2)}$ cyclic and elementary abelian of arbitrary order.

Let $G$ be a nilpotent group in which every coset $x[G, G]$ for $x \notin [G, G]$ coincides with the conjugacy class $x^G$. Is there a bound for the nilpotency class of $G$?

Does the free product of two groups with stable first-order theory also have stable first-order theory?

Is it consistent with ZFC that every non-discrete topological group contains a nonempty nowhere dense subset without isolated points?

Is it true that for every finite group $G$, there is a finite group $F$ and a surjective homomorphism $f : F \to G$ such that for each nontrivial subgroup $H$ of $F$, the restriction $f|_H$ is not injective?

It is known that for every finite group $G$ there is a finite group $F$ and a surjective homomorphism $f : F \to G$ such that for each subgroup $H$ of $F$ the restriction $f|_H$ is not bijective.

Is it true that every non-discrete topological group containing no countable open subgroup of exponent 2 can be partitioned into two (infinitely many) dense subsets?

(N. Eggert). Let $R$ be a commutative associative finite-dimensional nilpotent algebra over a finite field $F$ of characteristic $p$. Let $R^{(p)}$ be the subalgebra of all elements of the form $r^p$ ($r \in R$). Is it true that $\dim R \geqslant p \,\dim R^{(p)}$?

A finite group $G$ is called simply reducible (SR-group) if every element of $G$ is conjugate to its inverse, and the tensor product of any two irreducible representations of $G$ decomposes into a sum of irreducible representations of $G$ with coefficients 0 or 1.
$\qquad$ a) Is it true that the nilpotent length of a soluble SR-group is at most 5?
$\qquad$ b) Is it true that the derived length of a soluble SR-group is bounded by some constant $c$?

Does there exist a non-hereditary local formation $\mathfrak{F}$ of finite groups such that in every finite group the set of all $\mathfrak{F}$-subnormal subgroups is a sublattice of the subgroup lattice?

A negative answer would solve 9.75, since all the hereditary local formations with the same property are already known.

Does there exist an absolute constant $k$ such that for any prefrattini subgroup $H$ in any finite soluble group $G$ there exist $k$ conjugates of $H$ whose intersection is $\Phi(G)$?

Suppose that a subgroup $H$ of a finite group $G$ is such that $HM = MH$ for every minimal non-nilpotent subgroup $M$ of $G$. Must $H/H_G$ be nilpotent, where $H_G$ is the largest normal subgroup of $G$ contained in $H$?

Let $r(m) = \{r+km \mid k \in \mathbb{Z}\}$ for integers $0 \leqslant r < m$. For $r_1(m_1) \cap r_2(m_2) = \varnothing$ define the class transposition $\tau_{r_1(m_1), r_2(m_2)}$ as the involution which interchanges $r_1 + km_1$ and $r_2 + km_2$ for each integer $k$ and fixes everything else. The group $\text{CT}(\mathbb{Z})$ generated by all class transpositions is simple (Math. Z., 264, no. 4 (2010), 927–938). Is $\text{Out}(\text{CT}(\mathbb{Z})) = \langle \sigma \mapsto \sigma^{n \,\mapsto\, -n-1} \rangle \cong C_2$?

Does $\text{CT}(\mathbb{Z})$ (see 17.57 for definition) have subgroups of intermediate (word-) growth?

A permutation of $\mathbb{Z}$ is called residue-class-wise affine if there is a positive integer $m$ such that its restrictions to the residue classes (mod $m$) are all affine. Is $\text{CT}(\mathbb{Z})$ (see 17.57) the group of all residue-class-wise affine permutations of $\mathbb{Z}$ which fix the nonnegative integers setwise?

Given a set $\mathcal{P}$ of odd primes, let $\text{CT}_\mathcal{P}(\mathbb{Z})$ denote the subgroup of $\text{CT}(\mathbb{Z})$ (see 17.57) which is generated by all class transpositions which interchange residue classes whose moduli have only prime factors in $\mathcal{P} \cup \{2\}$. The groups $\text{CT}_\mathcal{P}(\mathbb{Z})$ are simple (Math. Z., 264, no. 4 (2010), 927–938). Are they pairwise nonisomorphic?

The group $\text{CT}_\mathcal{P}(\mathbb{Z})$ (see 17.60) is finitely generated if and only if $\mathcal{P}$ is finite. If $\mathcal{P} = \varnothing$, then it is isomorphic to the finitely presented (first) Higman–Thompson group (J. P. McDermott, see Remark 1.4 in S. Kohl, J. Group Theory, 20, no. 5 (2017), 1025–1030). Is it always finitely presented if $\mathcal{P}$ is finite?

Given a free group $F_n$ and a proper characteristic subgroup $C$, is it ever possible to generate the quotient $F_n/C$ by fewer than $n$ elements?

Prove that if $p$ is an odd prime and $s$ is a positive integer, then there are only finitely many $p$-adic space groups of finite coclass with point group of coclass $s$.

Say that a group $G$ is an $n$-approximation to the Nottingham group $J = N(p)$ (as defined in 12.24) if $G$ is an infinite pro-$p$ group, and $G/\gamma_n(G)$ is isomorphic to $J/\gamma_n(J)$. Does there exist a function $f(p)$ such that, if $G$ is an $f(p)$-approximation to the Nottingham group, then $\gamma_i(G)/\gamma_{i+1}(G)$ is isomorphic to $\gamma_i(J)/\gamma_{i+1}(J)$ for all $i$?

Cf. 14.56. Note that an affirmative solution to this problem trivially implies the now known fact that $J$ is finitely presented as a pro-$p$ group (if $p > 2$), see 14.55.

If Problem 17.64 has an affirmative answer, does it follow that, for some function $g(p)$, the number of isomorphism classes of $g(p)$-approximations to $J$ is
$\qquad$ a) countable?
$\qquad$ b) one?

Note that if, for some $g(p)$, there are only finitely many isomorphism classes, then for some $h(p)$ there is only one.

(R. Guralnick). Does there exist a positive integer $d$ such that $\dim H^1(G, V) \leqslant d$ for any faithful absolutely irreducible module $V$ for any finite group $G$? Cf. 16.55.

(H. Zassenhaus). Conjecture: Every invertible element of finite order of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group ring $\mathbb{Q}G$ to an element of $\pm G$.

Let $C$ be a cyclic subgroup of a group $G$ and $G = CFC$ where $F$ is a finite cyclic subgroup. Is it true that $|G:C|$ is finite? Cf. 12.62.

Let $G$ be a group of prime exponent acting freely on a non-trivial abelian group. Is $G$ cyclic?

Let $\alpha$ be an automorphism of prime order $q$ of an infinite free Burnside group $G = B(q, p)$ of prime exponent $p$ such that $\alpha$ cyclically permutes the free generators of $G$. Is it true that $\alpha$ fixes some non-trivial element of $G$?

Let $\alpha$ be a fixed-point-free automorphism of prime order $p$ of a periodic group $G$.
$\qquad$ a) Is it true that $G$ does not contain a non-trivial $p$-element?
$\qquad$ b) Suppose that $G$ does not contain a non-trivial $p$-element. Is $\alpha$ a splitting automorphism?

Let $AB$ be a Frobenius group with kernel $A$ and complement $B$. Suppose that $AB$ acts on a finite group $G$ so that $GA$ is also a Frobenius group with kernel $G$ and complement $A$.
$\qquad$ a) Is the nilpotency class of $G$ bounded in terms of $|B|$ and the class of $C_G(B)$?
$\qquad$ b) Is the exponent of $G$ bounded in terms of $|B|$ and the exponent of $C_G(B)$?

Let $G$ be a finite simple group of Lie type defined over a field of characteristic $p$, and $V$ an absolutely irreducible $G$-module over a field of the same characteristic. Is it true that in the following cases the split extension of $V$ by $G$ must contain an element whose order is distinct from the order of any element of $G$?
$\qquad$ a) $G = U_4(q)$;
$\qquad$ b) $G = S_{2n}(q)$, $n \geqslant 3$;
$\qquad$ c) $G = O_{2n+1}(q)$, $n \geqslant 3$;
$\qquad$ d) $G = O^+_{2n}(q)$, $n \geqslant 4$;
$\qquad$ e) $G = O^-_{2n}(q)$, $n \geqslant 4$;
$\qquad$ f) $G = {}^3D_4(q)$, $q \neq 2$;
$\qquad$ g) $G = E_6(q)$;
$\qquad$ h) $G = {}^2E_6(q)$;
$\qquad$ i) $G = E_7(q)$;
$\qquad$ j) $G = G_2(2^m)$.

Let $G$ be a finite simple group of Lie type defined over a field of characteristic $p$ whose Lie rank is at least three, and $V$ an absolutely irreducible $G$-module over a field of characteristic that does not divide $p$. It is true that the split extension of $V$ by $G$ must contain an element whose order is distinct from the order of any element of $G$? The case of $G = U_n(p^m)$ is of special interest.

Can the Monster $M$ act on a nontrivial finite 3-group $V$ so that all elements of $M$ of orders 41, 47, 59, 71 have no nontrivial fixed points in $V$?

Does there exist a finite group $G$, with $|G| > 2$, such that there is exactly one element in $G$ which is not a commutator?

Let $k$ be a positive integer such that there is an insoluble finite group with exactly $k$ conjugacy classes. Is it true that a finite group of maximal order with exactly $k$ conjugacy classes is insoluble?

Does there exist a finitely generated group without free subsemigroups generating a proper variety containing $\mathfrak{A}_p\mathfrak{A}$?

Does there exist a finitely generated torsion group of unbounded exponent generating a proper variety?

Let $[u, {}_n v] := [u, \underbrace{v, \dots, v}_n]$. Is the group $M_n = \langle x, y \mid x = [x, {}_n y], y = [y, {}_n x] \rangle$ infinite for every $n > 2$? (See also 11.18.)

Given normal subgroups $R_1, \dots, R_n$ of a group $G$, let $[[R_1, \dots, R_n]] := \prod ( [\bigcap_{i \in I} R_i, \bigcap_{j \in J} R_j] )$, where the product is over all $I \cup J = \{1, \dots, n\}$, $I \cap J = \varnothing$.

Let $G$ be a free group, and let $R_i = \langle r_i \rangle^G$ be the normal closures of elements $r_i \in G$. It is known (B. Hartley, Yu. Kuzmin, J. Pure Appl. Algebra, 74 (1991), 247–256) that the quotient $(R_1 \cap R_2)/[R_1, R_2]$ is a free abelian group. On the other hand, the quotient $(R_1 \cap \dots \cap R_n)/[[R_1, \dots, R_n]]$ has, in general, non-trivial torsion for $n \geqslant 4$. Is this quotient always torsion-free for $n = 3$? This is known to be true if $r_1, r_2, r_3$ are not proper powers in $G$.

Is it true that in every finitely presented group the intersection of the derived series has trivial abelianization?

Does there exist a group such that every central extension of it is residually nilpotent, but there exists a central extension of a central extension of it which is not residually nilpotent?

An associative algebra $A$ is said to be Calabi–Yau of dimension $d$ (for short, $\text{CY}_d$) if there is a natural isomorphism of $A$-bimodules $\text{Ext}^d_{A\text{-bimod}}(A, A \otimes A) \cong A$ and $\text{Ext}^n_{A\text{-bimod}}(A, A \otimes A) = 0$ for $n \neq d$. By Kontsevich’s theorem, the complex group algebra $\mathbb{C}G$ of the fundamental group $G$ of a 3-dimensional aspherical manifold is $\text{CY}_3$. Is every group with $\text{CY}_3$ complex group algebra residually finite?

Let $\mathfrak{V}$ be a variety of groups such that any free group in $\mathfrak{V}$ has torsion-free integral homology groups in all dimensions. Is it true that $\mathfrak{V}$ is abelian?

(Simplest questions related to the Whitehead asphericity conjecture).
Let $\langle x_1, x_2, x_3 \mid r_1, r_2, r_3 \rangle$ be a presentation of the trivial group.
$\qquad$ a) Prove that the group $\langle x_1, x_2, x_3 \mid r_1, r_2 \rangle$ is torsion-free.
$\qquad$ b) Let $F = F(x_1, x_2, x_3)$ and $R_i = \langle r_i \rangle^F$. Is it true that the group $F/[R_1, R_2]$ is residually soluble?

Construct a group of intermediate growth with finitely generated Schur multiplier.

Compute $K_0(\mathbb{F}_2G)$ and $K_1(\mathbb{F}_2G)$, where $G$ is the first Grigorchuk group, $\mathbb{F}_2G$ its group algebra over the field of two elements, and $K_0, K_1$ the zeroth and first $K$-functors.

By Bousfield’s theorem, the free pronilpotent completion of a non-cyclic free group has uncountable Schur multiplier. Is it true that the free prosolvable completion (that is, the inverse limit of quotients by the derived subgroups) of a non-cyclic free group has uncountable Schur multiplier?

(G. Baumslag). A group is parafree if it is residually nilpotent and has the same lower central quotients as a free group. Is it true that $H_2(G) = 0$ for any finitely generated parafree group $G$?

Let $d(X)$ denote the derived length of a group $X$.
$\qquad$ a) Does there exist an absolute constant $k$ such that $d(G) - d(M) \leqslant k$ for every finite soluble group $G$ and any maximal subgroup $M$?
$\qquad$ b) Find the minimum $k$ with this property.

What are the non-abelian composition factors of a finite non-soluble group all of whose maximal subgroups are Hall subgroups?

(Well-known problem). Let $G$ be a compact topological group which has elements of arbitrarily high orders. Must $G$ contain an element of infinite order?

(Well-known problem). Can the free product $\mathbb{Z} \ast G$ of the infinite cyclic group $\mathbb{Z}$ and a nontrivial group $G$ be the normal closure of a single element?

Let $G$ be a permutation group on the finite set $\Omega$. A partition $\rho$ of $\Omega$ is said to be $G$-regular if there exists a subset $S$ of $\Omega$ such that $S^g$ is a transversal of $\rho$ for all $g \in G$. The group $G$ is said to be synchronizing if $|\Omega| > 2$ and there are no non-trivial proper $G$-regular partitions on $\Omega$.
a) Are the following primitive groups of affine type synchronizing:
$2^p.\text{PSL}(2, 2p+1)$ where both $p$ and $2p+1$ are prime, $p \equiv 3 \pmod 4$ and $p > 23$?
$2^{101}.\text{He}$?
b) For which finite simple groups $S$ are the groups $S \times S$ acting on $S$ by $(g, h): x \to g^{-1} x h$ non-synchronizing?

Does there exist a variety of groups which contains only countably many subvarieties but in which there is an infinite properly descending chain of subvarieties?

Is every variety of groups of exponent 4 finitely based?

A variety of groups is said to be small if it contains only countably many non-isomorphic finitely generated groups.
$\hspace{0.8cm}$ a) Is it true that if $G$ is a finitely generated group and the variety $\text{Var}(G)$ it generates is small then $G$ satisfies the maximal condition on normal subgroups?
$\hspace{0.8cm}$ b) Is it true that a variety is small if and only if all its finitely generated groups have the Hopf property?

Consider the group $B = \langle a, b \mid (bab^{-1})a(bab^{-1})^{-1} = a^2 \rangle$ introduced by Baumslag in 1969. The same relation is satisfied by the functions $f(x) = 2x$ and $g(x) = 2^x$ under the operation of composition in the group of germs of monotonically increasing to $\infty$ continuous functions on $(0, \infty)$, where two functions are identified if they coincide for all sufficiently large arguments. Is the representation $a \to f, b \to g$ of the group $B$ faithful?

Conjecture: A finite group is not simple if it has an irreducible complex character of odd degree vanishing on a class of odd length.

If true, this implies the solvability of groups of odd order, so a proof independent of CFSG is of special interest.

According to P. Hall, a group $G$ is said to be homogeneous if every isomorphism of its finitely generated subgroups is induced by an automorphism of $G$. It is known (Higman–Neumann–Neumann) that every group is embeddable into a homogeneous one. Is the same true for the category of representations of groups? A representation $(V, G)$ is finitely generated if $G$ is a finitely generated group, and $V$ a finitely generated module over the group algebra of $G$.

We say that two subsets $A, B$ of an infinite group $G$ are separated if there exists an infinite subset $X$ of $G$ such that $1 \in X$, $X = X^{-1}$, and $XAX \cap B = \varnothing$. Is it true that any two disjoint subsets $A, B$ of an infinite group $G$ satisfying $|A| < |G|$, $|B| < |G|$ are separated? This is so if $A, B$ are finite, or $G$ is Abelian.

Does there exist a continuum of sets $\pi$ of primes for which every finite group possessing a Hall $\pi$-subgroup is a $D_\pi$-group?

Let $\Gamma$ be a finite non-oriented graph on the set of vertices $\{x_1, \dots, x_n\}$ and let
$$S_\Gamma = \langle x_1, \dots, x_n \mid x_i x_j = x_j x_i \iff (x_i, x_j) \in \Gamma; \mathfrak{A}^2 \rangle$$ be a presentation of a partially commutative metabelian group $S_\Gamma$ in the variety of all metabelian groups. Is the universal theory of the group $S_\Gamma$ decidable?

An equation over a pro-$p$-group $G$ is an expression $v(x) = 1$, where $v(x)$ is an element of the free pro-$p$-product of $G$ and a free pro-$p$-group with basis $\{x_1, \dots, x_n\}$; solutions are sought in the affine space $G^n$. Is it true that a free pro-$p$-group is equationally Noetherian, that is, for any $n$ every system of equations in $x_1, \dots, x_n$ over this group is equivalent to some finite subsystem of it?

Does $G = \text{SL}_2(\mathbb{C})$ contain a 2-generated free subgroup that is conjugate in $G$ to a proper subgroup of itself?

Is the group $\langle a, b, t \mid a^t = ab, b^t = ba \rangle$ linear?

If not, this would be an easy example of a non-linear hyperbolic group.

A non-trivial group word $w$ is uniformly elliptic in a class $\mathcal{C}$ if there is a function $f: \mathbb{N} \to \mathbb{N}$ such that the width of $w$ in every $d$-generator $\mathcal{C}$-group $G$ is bounded by $f(d)$ (i.e. every element of the verbal subgroup $w(G)$ is equal to a product of $f(d)$ values of $w$ or their inverses). If $\mathcal{C}$ is a class of finite groups, this is equivalent to saying that in every finitely generated pro-$\mathcal{C}$ group $G$ the verbal subgroup $w(G)$ is closed. A. Jaikin-Zapirain (Revista Mat. Iberoamericana, 24 (2008), 617–630) proved that $w$ is uniformly elliptic in finite $p$-groups if and only if $w \notin F''(F')^p$, where $F$ is the free group on the variables of $w$. Is it true that $w$ is uniformly elliptic in $\mathcal{C}$ if and only if $w \notin F''(F')^p$ for every prime $p$ in the case where $\mathcal{C}$ is the class of all
$\qquad$ a) finite soluble groups?
$\qquad$ b) finite groups?

See also Ch. 4 of (D. Segal, Words: notes on verbal width in groups, LMS Lect. Note Series, 361, Cambridge Univ. Press, 2009.

a) Is it true that for each word $w$, there is a function $h: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ such that the width of $w$ in every finite $p$-group of Prüfer rank $r$ is bounded by $h(p, r)$?
b) If so, can $h(p, r)$ be made independent of $p$?

Let $G$ be a finite group, and $p$ a prime divisor of $|G|$. Suppose that every maximal subgroup of a Sylow $p$-subgroup of $G$ has a $p$-soluble supplement in $G$. Must $G$ be $p$-soluble?

A subgroup $A$ of a group $G$ is said to be $G$-permutable in $G$ if for every subgroup $B$ of $G$ there exists an element $x \in G$ such that $AB^x = B^xA$. A subgroup $A$ is said to be hereditarily $G$-permutable in $G$ if $A$ is $E$-permutable in every subgroup $E$ of $G$ containing $A$. Which finite non-abelian simple groups $G$ possess
$\qquad$ a) a non-trivial $G$-permutable subgroup?
$\qquad$ b) a non-trivial hereditarily $G$-permutable subgroup?

Do there exist for $p > 3$ 2-generator finite $p$-groups with deficiency zero (see 8.12) of arbitrarily high nilpotency class?

Does every generalized free product (with amalgamation) of two non-trivial groups have maximal subgroups?

Can a locally free non-abelian group have non-trivial Frattini subgroup?

Let $n(G)$ be the maximum of positive integers $n$ such that the $n$-th direct power of a finite simple group $G$ is 2-generated. Is it true that $n(G) \geqslant \sqrt{|G|}$?

(Well-known problem). If groups $A$ and $B$ have decidable elementary theories $\text{Th}(A)$ and $\text{Th}(B)$, must $\text{Th}(A \ast B)$ be decidable?

Suppose that a finite $p$-group $G$ has a subgroup of exponent $p$ and of index $p$. Must $G$ also have a characteristic subgroup of exponent $p$ and of index bounded in terms of $p$?

Suppose that a finite soluble group $G$ admits a soluble group of automorphisms $A$ of coprime order such that $C_G(A)$ has rank $r$. Let $|A| $ be the product of $l$ not necessarily distinct primes. Is there a linear function $f$ such that $G/F_{f(l)}(G)$ has $(|A|, r)$-bounded rank, where $F_{f(l)}(G)$ is the $f(l)$-th Fitting subgroup?

Is a residually finite group all of whose subgroups of infinite index are finite necessarily cyclic-by-finite?

Let $G$ be a group whose set of elements is the real numbers and which is “nicely definable” (see below). Does $G$ being $\aleph_\omega$-free imply it being free if “nicely definable” means
$\qquad$ a) being $F_\sigma$?
$\qquad$ b) being Borel?
$\qquad$ c) being analytic?
$\qquad$ d) being projective $L[\mathbb{R}]$?

See https://arxiv.org/pdf/math/0212250.pdf for justification of the restrictions.

The same questions as 17.121 for an abelian group $G$.

Do there exist finite groups $G_1, G_2$ such that $\pi_e(G_1) = \pi_e(G_2)$, $h(\pi_e(G_1)) < \infty$, and each non-abelian composition factors of each of the groups $G_1, G_2$ is not isomorphic to a section of the other? For definitions see 13.63.

Is the set of finitely presented metabelian groups recursively enumerable?

Does every finite group $G$ contain a pair of conjugate elements $a, b$ such that $\pi(G) = \pi(\langle a, b \rangle)$? This is true for soluble groups.

Suppose that $G$ is a residually finite group satisfying the identity $[x, y]^n = 1$. Must $[G, G]$ be locally finite?

An equivalent question: Let $G$ be a finite soluble group satisfying the identity $[x, y]^n = 1$; is the Fitting height of $G$ bounded in terms of $n$? Cf. 13.34.

Suppose that a finite soluble group $G$ of derived length $d$ admits an elementary abelian $p$-group of automorphisms $A$ of order $p^n$ such that $C_G(A) = 1$. Must $G$ have a normal series of $n$-bounded length with nilpotent factors of $(p, n, d)$-bounded nilpotency class?

An affirmative answer would follow from an affirmative answer to 11.125.

Let $T$ be a finite $p$-group admitting an elementary abelian group of automorphisms $A$ of order $p^2$ such that in the semidirect product $P = TA$ every element of $P \setminus T$ has order $p$. Does it follow that $T$ is of exponent $p$?