Let $G$ be a finite non-abelian group, and $Z(G)$ its centre. One can associate a graph $\Gamma_G$ with $G$ as follows: take $G \setminus Z(G)$ as vertices of $\Gamma_G$ and join two vertices $x$ and $y$ if $xy \neq yx$. Let $H$ be a finite non-abelian group such that $\Gamma_G \cong \Gamma_H$.
$\qquad$ a) If $H$ is simple, is it true that $G \cong H$?
$\qquad$ b) If $H$ is nilpotent, is it true that $G$ is nilpotent?
$\qquad$ c) If $H$ is solvable, is it true that $G$ is solvable?

A group $G$ is subgroup-separable if for any subgroup $H \leqslant G$ and element $x \in G \setminus H$ there is a homomorphism to a finite group $f : G \to F$ such that $f(x) \notin f(H)$. Is it true that a finitely generated solvable group is locally subgroup-separable if and only if it does not contain a solvable Baumslag–Solitar group? Solvable Baumslag–Solitar groups are $BS(1, n) = \langle a, b \mid bab^{-1} = a^n \rangle$ for $n > 1$.
Background: It is known that a finitely generated solvable group is subgroup-separable if and only if it is polycyclic (R. C. Alperin, in: Groups–Korea '98 (Pusan), de Gruyter, Berlin, 2000, 1–5).

Is it true that if $G$ is a finite group with all conjugacy classes of distinct sizes, then $G \cong S_3$?

Let $G$ be a finite group with $C, D$ two nontrivial conjugacy classes such that $CD$ is also a conjugacy class. Can $G$ be a non-abelian simple group?

A group is said to be perfect if it coincides with its derived subgroup. Does there exist a perfect locally finite $p$-group
$\qquad$ a) all of whose proper subgroups are hypercentral?
$\qquad$ b) all of whose proper subgroups are solvable?

Can a perfect locally finite $p$-group be generated by a subset of bounded exponent
$\qquad$ a) if all of its proper subgroups are hypercentral?
$\qquad$ b) if all of its proper subgroups are solvable?

Is it true that the membership problem is undecidable for any semidirect product $F_n \rtimes F_n$ of non-abelian free groups $F_n$?

An affirmative answer would imply an answer to 6.24. Recall that the membership problem is decidable for $F_n$ (M. Hall, 1949) and undecidable for $F_n \times F_n$ (K. A. Mikhailova, 1958).

The width $w(G')$ of the derived subgroup $G'$ of a finite non-abelian group $G$ is the smallest positive integer $m$ such that every element of $G'$ is a product of $\leqslant m$ commutators. Is it true that the maximum value of the ratio $w(G')/|G|$ is $1/6$ (attained at the symmetric group $S_3$)?

An element $g$ of a free group $F_n$ on the free generators $x_1, \dots, x_n$ is called a palindrome with respect to these generators if the reduced word representing $g$ is the same when read from left to right or from right to left. The palindromic length of an element $w \in F_n$ is the smallest number of palindromes in $F_n$ whose product is $w$. Is there an algorithm for finding the palindromic length of a given element of $F_n$?

Is there an algorithm for finding the primitive length of a given element of $F_n$? The definition of a primitive element is given in 14.84; the primitive length is defined similarly to the palindromic length in 16.9.

Let $G$ be a finite $p$-group. Does there always exist a finite $p$-group $H$ such that $\Phi(H) \cong [G, G]$?

Given a finite $p$-group $G$, we define a $\Phi$-extension of $G$ as any finite $p$-group $H$ containing a normal subgroup $N$ of order $p$ such that $H/N \cong G$ and $N \leqslant \Phi(H)$. Is it true that for every finite $p$-group $G$ there exists an infinite sequence $G = G_1, G_2, \dots$ such that $G_{i+1}$ is a $\Phi$-extension of $G_i$ for all $i = 1, 2, \dots$?

Does there exist a finite $p$-group $G$ all of whose maximal subgroups $H$ are special, that is, satisfy $Z(H) = [H, H] = \Phi(H)$?

Let $G$ be a finite 2-group such that $\Omega_1(G) \leqslant Z(G)$. Is it true that the rank of $G/G^2$ is at most double the rank of $Z(G)$?

An element $g$ of a group $G$ is an Engel element if for every $h \in G$ there exists $k$ such that $[h, g, \dots, g] = 1$, where $g$ occurs $k$ times; if there is such $k$ independent of $h$, then $g$ is said to be boundedly Engel.
$\qquad$ a) (B. I. Plotkin). Does the set of boundedly Engel elements of a group form a subgroup?
$\qquad$ b) Does the set of boundedly Engel elements form a subgroup in a torsion-free group?
$\qquad$ c) The same question for right-ordered groups?
$\qquad$ d) The same question for linearly ordered groups?

a) Does the set of (not necessarily boundedly) Engel elements of a group without elements of order 2 form a subgroup?
b) The same question for torsion-free groups.
c) The same question for right-ordered groups.
d) The same question for linearly ordered groups.

Is it true that a non-abelian simple group cannot contain Engel elements other than the identity element?

Does there exist a linearly orderable soluble group of derived length exactly $n$ that has a single proper normal relatively convex subgroup
$\qquad$ a) for $n = 3$?
$\qquad$ b) for $n > 4$?

Such groups do exist for $n = 2$ and for $n = 4$.

Is the variety of lattice-ordered groups generated by nilpotent groups finitely based? (Here the variety is considered in the signature of group and lattice operations.)

Let $\mathfrak{M}$ be a quasivariety of groups. The dominion $\text{dom}_A^\mathfrak{M}(H)$ of a subgroup $H$ of a group $A$ (in $\mathfrak{M}$) is the set of all elements $a \in A$ such that for any two homomorphisms $f, g : A \to B \in \mathfrak{M}$, if $f, g$ coincide on $H$, then $f(a) = g(a)$. Suppose that the set $\{\text{dom}_A^\mathfrak{N}(H) \mid \mathfrak{N}$ is a quasivariety, $\mathfrak{N} \subseteq \mathfrak{M}\}$ forms a lattice with respect to set-theoretic inclusion. Can this lattice be modular and non-distributive?

Given a non-central matrix $\alpha \in \text{SL}_n(F)$ over a field $F$ for $n > 2$, is it true that every non-central matrix in $\text{SL}_n(F)$ is a product of $n$ matrices, each similar to $\alpha$?

(Well-known problem). Let $E_n(A)$ be the subgroup of $\text{GL}_n(A)$ generated by elementary matrices. Is $\text{SL}_2(A) = E_2(A)$ when $A = \mathbb{Z}[x, 1/x]$?

Is there, for some $n > 2$ and a ring $A$ with 1, a matrix in $E_n(A)$ that is nonscalar modulo any proper ideal and is not a commutator?

The spectrum of a finite group is the set of orders of its elements. Does there exist a finite group $G$ whose spectrum coincides with the spectrum of a finite simple exceptional group $L$ of Lie type, but $G$ is not isomorphic to $L$?

Do there exist three pairwise non-isomorphic finite non-abelian simple groups with the same spectrum (see 16.24)?

We say that the prime graphs of finite groups $G$ and $H$ coincide if the sets of primes dividing their orders are the same, $\pi(G) = \pi(H)$, and for any distinct $p, q \in \pi(G)$ there is an element of order $pq$ in $G$ if and only if there is such an element in $H$. Does there exist a positive integer $k$ such that there are no $k$ pairwise non-isomorphic finite non-abelian simple groups with the same graphs of primes? Conjecture: $k = 5$.

Suppose that a finite group $G$ has the same spectrum as an alternating group. Is it true that G has at most one non-abelian composition factor?

Let $G$ be a connected linear reductive algebraic group over a field of positive characteristic, $X$ a closed subset of $G$, and let $X^k = \{x_1 \dots x_k \mid x_i \in X\}$.
$\qquad$ a) Is it true that there always exists a positive integer $c = c(X) > 1$ such that $X^c$ is closed?
$\qquad$ b) If $X$ is a conjugacy class of $G$ such that $X^2$ contains an open subset of $G$, then is $X^2 = G$?

Which finite simple groups of Lie type $G$ have the following property: for every semisimple abelian subgroup $A$ and proper subgroup $H$ of $G$ there exists $x \in G$ such that $A^x \cap H = 1$?

Suppose that $A$ and $B$ are subgroups of a group $G$ and $G = AB$. Will $G$ have composition (principal) series if $A$ and $B$ have composition (respectively, principal) series?

Suppose that a group $G$ has a composition series and let $\mathfrak{F}(G)$ be the formation generated by $G$. Is the set of all subformations of $\mathfrak{F}(G)$ finite?

Suppose that a group $G$ has a composition series and let $\text{Fit}(G)$ be the Fitting class generated by $G$. Is the set of all Fitting subclasses of $\text{Fit}(G)$ finite? Cf. 14.31.

Suppose that a finite $p$-group $G$ has an abelian subgroup $A$ of order $p^n$. Does $G$ contain an abelian subgroup $B$ of order $p^n$ that is normal in $\langle B^G \rangle$
$\qquad$ a) if $p = 3$?
$\qquad$ b) if $p = 2$?
$\qquad$ c) If $p = 3$ and $A$ is elementary abelian, does $G$ contain an elementary abelian subgroup $B$ of order $3^n$ that is normal in $\langle B^G \rangle$?

Suppose that $G$ is a finitely generated group acting faithfully on a regular rooted tree by finite-state automorphisms. Is the conjugacy problem decidable for $G$? See the definitions in (R. I. Grigorchuk, V. V. Nekrashevich, V. I. Sushchanskiĭ, Proc. Steklov Inst. Math., 2000, no. 4 (231), 128–203).

Is every finitely presented soluble group nilpotent-by-nilpotent-by-finite?

(Well-known problem). We call a finite group rational if all of its ordinary characters are rational-valued. Is every Sylow 2-subgroup of a rational group also a rational group?

Let $G$ be a solvable rational finite group with an extra-special Sylow 2-subgroup. Is it true that either $G$ is a 2-nilpotent group, or there is a normal subgroup $E$ of $G$ such that $G/E$ is an extension of a normal 3-subgroup by an elementary abelian 2-group?

Let $G$ be a soluble group, and let $A$ and $B$ be periodic subgroups of $G$. Is it true that any subgroup of $G$ contained in the set $AB = \{ab \mid a \in A, b \in B\}$ is periodic? This is known to be true if $AB$ is a subgroup of $G$.

(J. E. Humphreys, D. N. Verma). Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and let $u = u(\mathfrak{g})$ be the restricted enveloping algebra of $\mathfrak{g}$. By a theorem of Curtis every irreducible restricted $u$-module (i.e. every irreducible restricted $\mathfrak{g}$-module) is the restriction to $\mathfrak{g}$ of a (rational) $G$-module. Is it also true that every projective indecomposable $u$-module is the restriction of a rational $G$-module? This is true if $p \geqslant 2h - 2$ (where $h$ is the Coxeter number of $G$) by results of Jantzen.

Let $\Delta$ be a subgroup of the automorphism group of a free pro-$p$ group of finite rank $F$ such that $\Delta$ is isomorphic (as a profinite group) to the group $\mathbb{Z}_p$ of $p$-adic integers. Is the subgroup of fixed points of $\Delta$ in $F$ finitely generated (as a profinite group)?

Let $F$ be a free pro-$p$ group of finite rank $n > 1$. Does $\text{Aut}\,F$ possess an open subgroup of finite cohomological dimension?

Is a topological Abelian group $(G, \tau)$ compact if every group topology $\tau' \subseteq \tau$ on $G$ is complete? (The answer is yes if every continuous homomorphic image of $(G, \tau)$ is complete.)

Is there a partition of the group $\bigoplus_{\omega_1} (\mathbb{Z}/3\mathbb{Z})$ into three subsets whose complements do not contain cosets modulo infinite subgroups? (There is a partition into two such subsets.)

Is there in ZFC a countable non-discrete topological group not containing discrete subsets with a single accumulation point? (Such a group is known to exist under Martin’s Axiom.)

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; it is minimal if no point may be removed. Let $b(G)$ be the maximum, over all permutation representations of the finite group $G$, of the maximum size of a minimal base for $G$. Let $\mu'(G)$ be the maximum size of an independent set in $G$, a set of elements with the property that no element belongs to the subgroup generated by the others. Is it true that $b(G) = \mu'(G)$? (It is known that $b(G) \leqslant \mu'(G)$, and that equality holds for the symmetric groups.)

Remark. An equivalent question is the following. Suppose that the Boolean lattice $B(n)$ of subsets of an $n$-element set is embeddable as a meet-semilattice of the subgroup lattice of $G$, and suppose that $n$ is maximal with this property. Is it true that then there is such an embedding of $B(n)$ with the property that the least element of $B(n)$ is a normal subgroup of $G$?

Among the finitely-presented groups that act arc-transitively on the (infinite) 3-valent tree with finite vertex-stabilizer are the two groups
$$G_3 = \langle h, a, P, Q \mid h^3, a^2, P^2, Q^2, [P, Q], [h, P], (hQ)^2, a^{-1}PaQ \rangle$$ and
$$G_4 = \langle h, a, p, q, r \mid h^3, a^2, p^2, q^2, r^2, [p, q], [p, r], p(qr)^2, h^{-1}phq, h^{-1}qhpq, (hr)^2, [a, p], a^{-1}qar \rangle,$$ each of which contains the modular group $G_1 = \langle h, a \mid h^3, a^2 \rangle \cong \text{PSL}_2(\mathbb{Z})$ as a subgroup of finite index. The free product of $G_3$ and $G_4$ with subgroup $G_1 = \langle h, a \rangle$ amalgamated has a normal subgroup $K$ of index 8 generated by $A = h$, $B = aha$, $C = p$, $D = PpP$, $E = QpQ$, and $F = PQpQP$, with dihedral complement $\langle a, P, Q \rangle$. The group $K$ has presentation $$\begin{align}
\langle A, B, C, D, E, F \mid &{} A^3, B^3, C^2, D^2, E^2, F^2, (AC)^3, (AD)^3, (AE)^3, (AF)^3, (BC)^3, (BD)^3, (BE)^3, (BF)^3,\\
& (ABA^{-1}C)^2, (ABA^{-1}D)^2,
(A^{-1}BAE)^2, (A^{-1}BAF)^2, (BAB^{-1}C)^2, (B^{-1}ABD)^2,
(BAB^{-1}E)^2, (B^{-1}ABF)^2 \rangle.
\end{align}$$ Does this group have a non-trivial finite quotient?

(P. Conrad). Is it true that every torsion-free abelian group admits an Archimedean lattice ordering?

A group $H$ is said to have generalized torsion if there exists an element $h \neq 1$ such that $h^{x_1} h^{x_2} \dots h^{x_n} = 1$ for some $n$ and some $x_i \in H$. Is every group without generalized torsion right-orderable?

Is it true that a free product of groups without generalized torsion is a group without generalized torsion?

Do there exist simple finitely generated right-orderable groups?

Do there exist groups that can be right-ordered in infinitely countably many ways?

Is every finitely presented elementary amenable group solvable-by-finite?

Let $d(G)$ denote the smallest cardinality of a generating set of the group $G$. Suppose that $G = \langle A, B \rangle$, where $A$ and $B$ are two $d$-generated finite groups of coprime orders. Is it true that $d(G) \leqslant d + 1$? See 12.71 and (A. Lucchini, J. Algebra, 245 (2001), 552–561).

We say that a group $G$ acts freely on a group $V$ if $vg \neq v$ for any nontrivial elements $g \in G$, $v \in V$. Is it true that a group $G$ that can act freely on a non-trivial abelian group is embeddable in the multiplicative group of some skew-field?

(Well-known problem). Let $V$ be a faithful absolutely irreducible module for a finite group $G$. Is it true that $\dim H^1(G, V) \leqslant 2$?

The spectrum $\omega(G)$ of a group $G$ is the set of orders of elements of $G$. Suppose that $\omega(G) = \{1, 2, 3, 4, 5, 6\}$. Is $G$ locally finite?

Suppose that $\omega(G) = \omega(L_2(7)) = \{1, 2, 3, 4, 7\}$ (see 16.57 for notation). Is $G \cong L_2(7)$? This is true for finite $G$.

Is $SU_2(\mathbb{C})$ the only group that has just one irreducible complex representation of dimension $n$ for each $n = 1, 2, \dots$?

(If $R[n]$ is the $n$-dimensional irreducible complex representation of $SU_2(\mathbb{C})$, then $R[2]$ is the natural two-dimensional representation, and $R[2] \otimes R[n] = R[n-1] + R[n+1]$ for $n > 1$.)

Given a finite group $K$, does there exist a finite group $G$ such that $K \cong \text{Out } G = \text{Aut } G / \text{Inn } G$? (It is known that an infinite group $G$ exists with this property.)

If $G$ is a finite group, let $T(G)$ be the sum of the degrees of the irreducible complex representations of $G$, $T(G) = \sum_{i=1}^{k(G)} d_i$, where $G$ has $k(G)$ conjugacy classes. If $\alpha \in \text{Aut}\,G$, let $S_\alpha = \{g \in G \mid \alpha(g) = g^{-1}\}$. Is it true that $T(G) \geqslant |S_\alpha|$ for all $\alpha \in \text{Aut}\,G$?

A subgroup $H$ of a group $G$ is fully invariant if $\vartheta(H) \leqslant H$ for every endomorphism $\vartheta$ of $G$. Let $G$ be a finite group such that $G$ has a fully invariant subgroup of order $d$ for every $d$ dividing $|G|$. Must $G$ be cyclic?

Let $G$ be a group such that every $\alpha \in \text{Aut } G$ fixes every conjugacy class of $G$ (setwise). Must $\text{Aut } G = \text{Inn } G$?

Is there a non-trivial finite $p$-group $G$ of odd order such that $|\text{Aut}\,G| = |G|$? See also 12.77.

A non-abelian variety in which all finite groups are abelian is called pseudo-abelian. A group variety is called a $t$-variety if for all groups in this variety the relation of being a normal subgroup is transitive. By (O. Macedońska, A. Storozhev, Commun. Algebra, 25, no. 5 (1997), 1589–1593) each non-abelian $t$-variety is pseudo-abelian, and the pseudo-abelian varieties constructed in (A. Yu. Olshanskii, Math. USSR–Sb., 54 (1986), 57–80) are $t$-varieties. Is every pseudo-abelian variety a $t$-variety?

Does there exist a finitely presented residually torsion-free nilpotent group with a free presentation $G = F/R$ such that the group $F/[F, R]$ is not residually nilpotent?

For a group $G$, let $D_n(G)$ denote the $n$-th dimension subgroup of $G$, and $\zeta_n(G)$ the $n$-th term of its upper central series. For a given integer $n \geqslant 1$, let $f(n) = \max\{m \mid \exists$ a nilpotent group $G$ of class $n$ with $D_m(G) \neq 1\}$ and $g(n) = \max\{m \mid \exists$ a nilpotent group $G$ of class $n$ such that $D_n(G) \not\subseteq \zeta_m(G)\}$.
$\qquad$ a) What is $f(3)$?
$\qquad$ b) (B. I. Plotkin). Is it true that $f(n)$ is finite for all $n$?
$\qquad$ c) Is the growth of $f(n)$ and $g(n)$ polynomial, exponential, or intermediary?

Conjecture: Given any integer $k$, there exists an integer $n_0 = n_0(k)$ such that if $n \geqslant n_0$ then the symmetric group of degree $n$ has at least $k$ different ordinary irreducible characters of equal degrees.

Let $W(x, y)$ be a non-trivial reduced group word, and $G$ one of the groups $\text{PSL}(2, \mathbb{R})$, $\text{PSL}(2, \mathbb{C})$, or $\text{SO}(3, \mathbb{R})$. Are all the maps $W : G \times G \to G$ surjective?

Let $W(x, y)$ be a non-trivial reduced group word, considered as a map $W: G\times G\to G$, where $G=\text{GL}(2, \mathbb{R})$.
$\qquad$ a) Must the range of the function $\text{Tr}(W(x, y))$ for $x, y \in \text{GL}(2, \mathbb{R})$ include the interval $[-2, +\infty)$?
$\qquad$ b) For $x, y$ being non-zero quaternions, must the range of the function $\text{Re}(W(x, y))$ include the interval $[-5/27, 1]$?

Suppose that a finitely generated group $G$ acts freely on an $\Lambda$-tree, where $\Lambda$ is an ordered abelian group. Is it true that $G$ acts freely on a $\mathbb{Z}^n$-tree for some $n$?

Is the elementary theory of a torsion-free hyperbolic group decidable?

Does there exist an exponential-time algorithm for obtaining a JSJ-decomposition of a finitely generated fully residually free group?

Let $G$ be a group generated by a finite set $S$, and let $l(g)$ denote the word length function of $g \in G$ with respect to $S$. The group $G$ is said to be contracting if there exist a faithful action of $G$ on the set $X^*$ of finite words over a finite alphabet $X$ and constants $0 < \lambda < 1$ and $C > 0$ such that for every $g \in G$ and $x \in X$ there exist $h \in G$ and $y \in X$ such that $l(h) < \lambda l(g) + C$ and $g(xw) = yh(w)$ for all $w \in X^*$.
$\qquad$ a) Can a contracting group have a non-abelian free subgroup?
$\qquad$ b) Do there exist non-amenable contracting groups?

a) Let $G = \langle \alpha, \beta \rangle$ be the group generated by the following two permutations of $\mathbb{Z}$: $\alpha(n) = n + 1$; $\beta(0) = 0$, $\beta(2^k m) = 2^k(m + 2)$, where $m$ is odd and $k$ is a positive integer. Is $G$ amenable?
b) Is it true that all groups generated by automata of polynomial growth in the sense of S. Sidki (Geom. Dedicata, 108 (2004), 193–204) are amenable?

Can a non-abelian one-relator group be the group of all automorphisms of some group?

We call a group $G$ strictly real if each of its non-trivial elements is conjugate to its inverse by some involution in $G$. In which groups of Lie type over a field of characteristic 2 the maximal unipotent subgroups are strictly real?

It is known that in every Noetherian group the nilpotent radical coincides with the collection of all Engel elements (R. Baer, Math. Ann., 133 (1957), 256–270; B. I. Plotkin, Izv. Vyssh. Uchebn. Zaved. Mat., 1958, no. 1(2), 130–135 (Russian)). It would be nice to find a similar characterization of the solvable radical of a finite group. More precisely, let $u = u(x, y)$ be a sequence of words satisfying 15.75. We say that an element $g \in G$ is $u$-Engel if there exists $n = n(g)$ such that $u_n(x, g) = 1$ for every element $x \in G$. Does there exist a sequence $u = u(x, y)$ such that the solvable radical of a finite group coincides with the set of all $u$-Engel elements?

Do there exist linear non-abelian simple groups without involutions?

Is it true that in any finitely generated $AT$-group over a sequence of cyclic groups of uniformly bounded orders all Sylow subgroups are locally finite? For the definition of an $AT$-group see (A. V. Rozhkov, Math. Notes, 40 (1986), 827–836)

Suppose that a group $G$ is obtained from the free product of torsion-free groups $A_1, \dots, A_n$ by imposing $m$ additional relations, where $m < n$. Is it true that the free product of some $n - m$ of the $A_i$ embeds into $G$?

Let $\mathcal{X}$ be a non-empty class of finite groups closed under taking homomorphic images, subgroups, and direct products. With every group $G \in \mathcal{X}$ we associate some set $\tau(G)$ of subgroups of $G$. We say that $\tau$ is a subgroup functor on $\mathcal{X}$ if:
$\qquad$ 1) $G \in \tau(G)$ for all $G \in \mathcal{X}$, and
$\qquad$ 2) for each epimorphism $\varphi : A \to B$, where $A, B \in \mathcal{X}$, and for any $H \in \tau(A)$ and $T \in \tau(B)$ we have $H^\varphi \in \tau(B)$ and $T^{\varphi^{-1}} \in \tau(A)$.

A subgroup functor $\tau$ is closed if for each group $G \in \mathcal{X}$ and for every subgroup $H \in \mathcal{X} \cap \tau(G)$ we have $\tau(H) \subseteq \tau(G)$. The set $\mathcal{F}(\mathcal{X})$ consisting of all closed subgroup functors on $\mathcal{X}$ is a lattice (in which $\tau_1 \leqslant \tau_2$ if and only if $\tau_1(G) \subseteq \tau_2(G)$ for every group $G \in \mathcal{X}$). It is known that $\mathcal{F}(\mathcal{X})$ is a chain if and only if $\mathcal{X}$ is a class of $p$-groups for some prime $p$ (Theorem 1.5.17 in S. F. Kamornikov and M. V. Sel'kin, Subgroups functors and classes of finite groups, Belaruskaya Navuka, Minsk, 2001 (Russian)).

Is there a non-nilpotent class $\mathcal{X}$ such that the width of the lattice $\mathcal{F}(\mathcal{X})$ is at most $|\pi(\mathcal{X})|$ where $\pi(\mathcal{X})$ is the set of all prime divisors of the orders of the groups in $\mathcal{X}$?

Let $E_n$ be a free locally nilpotent $n$-Engel group on countably many generators, and let $\pi(E_n)$ be the set of prime divisors of the orders of elements of the periodic part of $E_n$. It is known that $2, 3, 5 \in \pi(E_4)$.
$\qquad$ a) Does there exist $n$ for which $7 \in \pi(E_n)$?
$\qquad$ b) Is it true that $\pi(E_n) = \pi(E_{n+1})$ for all sufficiently large $n$?

Can the braid group $B_n$, $n \geqslant 4$, act faithfully on a regular rooted tree by finite-state automorphisms? Such action is known for $B_3$.

See the definitions in (R. I. Grigorchuk, V. V. Nekrashevich, V. I. Sushchanskiĭ, Proc. Steklov Inst. Math., 2000, no. 4 (231), 128–203).

Suppose that groups $G, H$ act faithfully on a regular rooted tree by finite-state automorphisms. Can their free product $G * H$ act faithfully on a regular rooted tree by finite state automorphisms?

See the definitions in (R. I. Grigorchuk, V. V. Nekrashevich, V. I. Sushchanskiĭ, Proc. Steklov Inst. Math., 2000, no. 4 (231), 128–203).

Does the group of all finite-state automorphisms of a regular rooted tree possess an irreducible system of generators?

Let $\mathfrak{M}$ be a variety of groups and let $G_r$ be a free $r$-generator group in $\mathfrak{M}$. A subset $S \subseteq G_r$ is called a test set if every endomorphism of $G_r$ identical on $S$ is an automorphism. The minimum of the cardinalities of test sets is called the test rank of $G_r$. Suppose that the test rank of $G_r$ is $r$ for every $r \geqslant 1$.
$\qquad$ a) Is it true that $\mathfrak{M}$ is an abelian variety?
$\qquad$ b) Suppose that $\mathfrak{M}$ is not a periodic variety. Is it true that $\mathfrak{M}$ is the variety of all abelian groups?

(G. M. Bergman). The width of a group $G$ with respect to a generating set $X$ means the supremum over all $g \in G$ of the least length of a group word of $X$ expressing $g$. A group $G$ has finite width if the width of $G$ with respect to every generating set is finite. Does there exist a countably infinite group of finite width?

All known infinite groups of finite width (infinite permutation groups, infinite-dimensional general linear groups, and some other groups) are uncountable (G. M. Bergman, Bull. London Math. Soc., 38 (2006), 429–440, and references therein).

(G. M. Bergman). Is it true that the automorphism group of an infinitely generated free group $F$ has finite width? The answer is affirmative if $F$ is countably generated.

Is it true that the automorphism group $\text{Aut}\,F$ of an infinitely generated free group $F$ is
$\qquad$ a) the normal closure of a single element?
$\qquad$ b) the normal closure of some involution in $\text{Aut}\,F$?

Let $F$ be an infinitely generated free group. Is there an $\text{IA}$-automorphism of $F$ whose normal closure in $\text{Aut}\,F$ is the group of all $\text{IA}$-automorphisms of $F$?

Let $F$ be an infinitely generated free group. Is $\text{Aut}\,F$ equal to its derived subgroup? This is true if $F$ is countably generated (R. Bryant, V. A. Roman’kov, J. Algebra, 209 (1998), 713–723).

Let $F_n$ be a free group of finite rank $n \geqslant 2$. Is the group $\text{Inn}\,F_n$ of inner automorphisms of $F_n$ a first-order definable subgroup of $\text{Aut}\,F_n$? It is known that the set of inner automorphisms induced by powers of primitive elements is definable in $\text{Aut}\,F_n$.

If $G = [G, G]$, then the commutator width of the group $G$ is its width relative to the set of commutators. Let $V$ be an infinite-dimensional vector space over a division ring. It is known that the commutator width of $\text{GL}(V)$ is finite. Is it true that the commutator width of $\text{GL}(V)$ is one?

Conjecture: If $F$ is a field and $A$ is in $\text{GL}(n, F)$, then there is a permutation matrix $P$ such that $AP$ is cyclic, that is, the minimal polynomial of $AP$ is also its characteristic polynomial.

Let $G$ be a locally finite $n$-Engel $p$-group where $p$ is a prime greater than $n$. Is $G$ then a Fitting group? (Examples of N. Gupta and F. Levin show that the condition $p > n$ is necessary in general.)

Let $G$ be a torsion-free group with all subgroups subnormal of defect at most $n$. Must $G$ then be nilpotent of class at most $n$? (This is known to be true for $n < 5$).

Suppose that $G$ is a solvable finite group and $A$ is a group of automorphisms of $G$ of relatively prime order. Is there a bound for the Fitting height $h(G)$ of $G$ in terms of $A$ and $h(C_G(A))$, or even in terms of the length $l(A)$ of the longest chain of nested subgroups of $A$ and $h(C_G(A))$?

Suppose that $G$ is a finite solvable group, $A \leqslant \text{Aut}\,G$, $C_G(A) = 1$, the orders of $G$ and $A$ are coprime, and let $l(A)$ be the length of the longest chain of nested subgroups of $A$. Is the Fitting height of $G$ bounded above by $l(A)$?

Is there an (infinite) 2-generator simple group $G$ such that $\text{Aut}\,F_2$ is transitive on the set of normal subgroups $N$ of the free group $F_2$ such that $F_2/N \cong G$? Cf. 6.45.

Do there exist uncountably many infinite 2-groups that are quotients of the group $\langle x, y \mid x^2 = y^4 = (xy)^8 = 1 \rangle$? There certainly exists one, namely the subgroup of finite index in Grigorchuk's first group generated by $b$ and $ad$; see (R. I. Grigorchuk, Functional Anal. Appl., 14 (1980), 41–43).

We say that a group $G$ is rational if any two elements $x, y \in G$ satisfying $\langle x \rangle = \langle y \rangle$ are conjugate. Is it true that for any $d$ there exist only finitely many finite rational $d$-generated 2-groups?

Is there a rank analogue of the Leedham-Green–McKay–Shepherd theorem on $p$-groups of maximal class? More precisely, suppose that $P$ is a 2-generator finite $p$-group whose lower central quotients $\gamma_i(P)/\gamma_{i+1}(P)$ are cyclic for all $i \geqslant 2$. Is it true that $P$ contains a normal subgroup $N$ of nilpotency class $\leqslant 2$ such that the rank of $P/N$ is bounded in terms of $p$ only?

If $G$ is a finite group, then every element $a$ of the rational group algebra $\mathbb{Q}[G]$ has a unique Jordan decomposition $a = a_s + a_n$, where $a_n \in \mathbb{Q}[G]$ is nilpotent, $a_s \in \mathbb{Q}[G]$ is semisimple over $\mathbb{Q}$, and $a_s a_n = a_n a_s$. The integral group ring $\mathbb{Z}[G]$ is said to have the additive Jordan decomposition property (AJD) if $a_s, a_n \in \mathbb{Z}[G]$ for every $a \in \mathbb{Z}[G]$. If $a \in \mathbb{Q}[G]$ is invertible, then $a_s$ is also invertible and so $a = a_s a_u$ with $a_u = 1 + a_s^{-1}a_n$ unipotent and $a_s a_u = a_u a_s$. Such a decomposition is again unique. We say that $\mathbb{Z}[G]$ has multiplicative Jordan decomposition property (MJD) if $a_s, a_u \in \mathbb{Z}[G]$ for every invertible $a \in \mathbb{Z}[G]$. See the survey (A. W. Hales, I. B. S. Passi, in: Algebra, Some Recent Advances, Birkhäuser, Basel, 1999, 75–87).

Is it true that there are only finitely many isomorphism classes of finite 2-groups $G$ such that $\mathbb{Z}[G]$ has MJD but not AJD?

Is it true that a locally graded group which is a product of two almost polycyclic subgroups (equivalently, of two almost soluble subgroups with the maximal condition) is almost polycyclic?

Let $\pi_e(G)$ denote the set of orders of elements of a group $G$, and $h(\Gamma)$ the number of non-isomorphic finite groups $G$ with $\pi_e(G) = \Gamma$. Do there exist two finite groups $G_1, G_2$ such that $\pi_e(G_1) = \pi_e(G_2)$, $h(\pi_e(G_1)) < \infty$, and neither of the two groups $G_1, G_2$ is isomorphic to a subgroup or a quotient of a normal subgroup of the other?

Is it true that almost every alternating group $A_n$ is uniquely determined in the class of finite groups by its set of element orders, i. e. that $h(\pi_e(A_n)) = 1$ for all large enough $n$?

Do braid groups $B_n$, $n > 4$, have non-elementary hyperbolic factor groups?

Is there a polynomial time algorithm for solving the word problem in the group $\text{Aut } F_n$ (with respect to some particular finite presentation), where $F_n$ is the free group of rank $n \geqslant 2$?

(I. Kapovich, P. Schupp). Is there an algorithm which, when given two elements $u, v$ of a free group $F_n$, decides whether or not the cyclic length of $\phi(u)$ equals the cyclic length of $\phi(v)$ for every automorphism $\phi$ of $F_n$?

Must an infinite simple periodic group with a dihedral Sylow 2-subgroup be isomorphic to $L_2(P)$ for a locally finite field $P$ of odd characteristic?