For $k \geqslant 1$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\text{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits in the induced action on the set of ordered $k$-tuples $\Omega^k$. Are there any finite insoluble totally 2-closed groups with nontrivial Fitting subgroup?

The finite totally 2-closed groups which are either soluble or have trivial Fitting subgroup are known.

Are there any nonabelian simple groups of Lie type which are totally 3-closed (see 20.1)?

There are exactly six nonabelian simple totally 2-closed groups – all are sporadic groups, the largest being the Monster.

For each $k$, each group of order $k$ is totally $k$-closed, while the alternating group $A_k$ is not totally $(k-2)$-closed (see 20.1 for definition). Is there a fixed integer $k$ such that all finite simple groups which are not alternating groups are totally $k$-closed?

A finite group $G$ is said to be cut (or inverse semi-rational) if $\langle x \rangle = \langle y \rangle$ implies that $x$ is conjugate to $y$ or to $y^{-1}$ for all $x, y \in G$.
$\qquad$ a) Let $\mathbb{Q}(G)$ denote the field extension of the rationals obtained by adjoining all entries of the ordinary character table of $G$. Is there $c > 0$ such that $|\mathbb{Q}(G) : \mathbb{Q}| \leqslant c$ for all cut groups? This is true if one assumes in addition that $G$ is solvable (J. F. Tent, J. Algebra, 363 (2012), 73–82).
$\qquad$ b) Is a Sylow 3-subgroup of a cut group also a cut group?
$\qquad$ c) Let $O_p(G)$ denote the largest normal $p$-subgroup of $G$. Let $G$ be a solvable cut group. Is it true that for $p \in \{5, 7\}$ the exponent of $O_p(G)$ divides $p$?

For additional information and some positive results see (Adv. Group Theory Appl., 8B, 2020, 157–160 or https://arxiv.org/abs/2001.02637).

(Well-known problem). Let $G$ be a finite group, and $V(\mathbb{Z}G)$ the group of normalized units of the integral group ring of $G$. Do the spectra of $G$ and $V(\mathbb{Z}G)$ coincide? That is, is it true that, for any integer $n$, there is an element of order $n$ in $V(\mathbb{Z}G)$ if and only if there is an element of order $n$ in $G$?

(W. Kimmerle). The prime graph (or Gruenberg–Kegel graph) $\Gamma(X)$ of a group $X$ has vertices labeled by primes appearing as orders of elements in $X$; two distinct primes $p$ and $q$ are adjacent in $\Gamma(X)$ if and only if $X$ contains an element of order $pq$. Denote by $V(\mathbb{Z}G)$ the group of normalized units of the integral group ring of a group $G$. Is it true that for each finite group $G$ the prime graphs of $G$ and $V(\mathbb{Z}G)$ coincide?

(W. Boone and G. Higman) Does every finitely generated group with solvable word problem embed into a finitely presented simple group?

It is known that every such group embeds into a simple subgroup of a finitely presented group (W. Boone, G. Higman, J. Austral. Math. Soc., 18, no. 1 (1974), 41–53).

Suppose $K < H < F$ are free groups of finite rank such that $\text{rank}(H) < \text{rank}(K)$, but all proper subgroups of $H$ which contain $K$ have ranks $\geqslant \text{rank}(K)$. Then is the inclusion of $H$ in $F$ the only homomorphism $H \to F$ fixing all elements of $K$?

In the group algebra of a free group over a field, does every element whose support in the group has cardinality more than 1 generate a proper 2-sided ideal?

This question is one of several related questions in (G. M. Bergman, Commun. Algebra, 49, no. 9 (2021), 3760–3776).

(a) If $\mathscr{U}$ and $\mathscr{U}'$ are nonprincipal ultrafilters on $\mathbb{N}$, can every group which can be written as a homomorphic image of an ultraproduct of groups with respect to $\mathscr{U}$ also be written as a homomorphic image of an ultraproduct of groups with respect to $\mathscr{U}'$ ?

(b) If the answer to (a) is negative, is it at least true that for any two nonprincipal ultrafilters $\mathscr{U}$ and $\mathscr{U}'$ on $\mathbb{N}$, there exists a nonprincipal ultrafilter $\mathscr{U}''$ on $\mathbb{N}$ such that every group which can be written as a homomorphic image of an ultraproduct of groups with respect to $\mathscr{U}$ or with respect to $\mathscr{U}'$ can be written as a homomorphic image of an ultraproduct with respect to $\mathscr{U}''$?

A positive answer to (b) would imply that the class of groups which can be written as homomorphic images of nonprincipal countable ultraproducts of groups is closed under finite direct products. See (G. M. Bergman, Pacific J. Math., 274 (2015), 451–495).

Let $F \leqslant H$ be free groups such that there exists a free group $G$ of finite rank with $F \leqslant G \leqslant H$, and let $r$ be the least of the ranks of such groups $G$. Which, if any, of the following statements must hold?
$\qquad$ (i) There is a largest $G$ of rank $r$ between $F$ and $H$.
$\qquad$ (i$'\,$) For any two $G_1$, $G_2$ of rank $r$ between $F$ and $H$, the subgroup $\langle G_1, G_2 \rangle$ has rank $r$.
$\qquad$ (ii) There is a smallest $G$ of rank $r$ between $F$ and $H$.
$\qquad$ (ii$'\,$) For any two $G_1, G_2$ of rank $r$ between $F$ and $H$, the subgroup $G_1 \cap G_2$ has rank $r$.

If (i) holds for all such $F$ and $H$, then so does (i$'\,$). If (ii) holds for all $F, H$, then so does (ii$'\,$). The converse of the former implication holds because subgroups of a free group of any fixed finite rank satisfy ACC, but I don’t see a way to get the converse of the other implication.

Let us say that a group $G$ has the unique $n$-fold product property (u.-n-p.) if for every $n$-tuple of finite nonempty subsets $A_1, \dots, A_n \subseteq G$ there exists $g \in G$ which can be written in one and only one way as $g = a_1 \dots a_n$ with $a_i \in A_i$. It is easy to see that u.-n-p. implies u.-m-p. for $n \geqslant m$ (since some of the $A_i$ can be $\{1\}$).

Are the conditions u.-n-p. ($n \geqslant 2$) all equivalent to u.-2-p., the usual unique product condition?

(a) If an abelian group can be written as a homomorphic image of a nonprincipal countable ultraproduct of not necessarily abelian groups $G_i$, must it be a homomorphic image of a nonprincipal countable ultraproduct of abelian groups? See (G. M. Bergman, Pacific J. Math., 274 (2015), 451–495).

(b) If an abelian group can be written as a homomorphic image of a direct product of an infinite family of not necessarily abelian finite groups, can it be written as a homomorphic image of a direct product of finite abelian groups?

To see that neither question is trivial, choose for each $n > 0$ a finite group $G_n$ which is perfect but has elements which cannot be written as products of fewer than $n$ commutators. Then both the direct product of the $G_n$ and any nonprincipal ultraproduct of those groups will have elements which are not products of commutators; hence its abelianization $A$ will be nontrivial. There is no evident family of abelian groups from which to obtain $A$ as an image of a nonprincipal ultraproduct, nor a family of finite abelian groups from which to obtain $A$ as an image of a direct product.

(a) Do there exist a variety $\mathfrak{V}$ of groups and a group $G \in \mathfrak{V}$ such that the coproduct in $\mathfrak{V}$ of two copies of $G$ is embeddable in $G$, but the coproduct of three such copies is not? See (G. M. Bergman, Indag. Math., 18 (2007), 349–403).

Given an embedding $G \ast_{\mathfrak{V}} G \to G$, one might expect the induced map
$$G \ast_{\mathfrak{V}} (G \ast_{\mathfrak{V}} G) \to G \ast_{\mathfrak{V}} G \to G$$ to be an embedding. But this is not automatic, because in a general group variety $\mathfrak{V}$, a map $G \ast_{\mathfrak{V}} A \to G \ast_{\mathfrak{V}} B$ induced by an embedding $A \to B$ is not necessarily again an embedding.

(b) If there exist $\mathfrak{V}$ and $G$ as in (a), does there in fact exist an example with $\mathfrak{V}$ the variety of groups generated by $G$? For this and related questions, see (G. M. Bergman, Algebra Number Theory, 3 (2009), 847–879).

Let $\kappa$ be an infinite cardinal. If a residually finite group $G$ is embeddable in the full permutation group of a set of cardinality $\kappa$, must it be embeddable in the direct product of $\kappa$ finite groups? See (G. M. Bergman, Indag. Math., 18 (2007), 349–403).

The converse is true: any such direct product is residually finite and embeddable in the indicated permutation group. Also, the statement asked for becomes true if one replaces “direct product of $\kappa$ finite groups” by “direct product of $2^\kappa$ finite groups”, since $G$ has cardinality at most $2^\kappa$, hence homomorphisms to that many finite groups can be chosen which, together, separate each element of $G$ from $e$.

By Hartley’s theorem (based on CFSG), if a simple locally finite group does not contain a particular finite group as a section, then it is isomorphic to a group of Lie type over a locally finite field.
$\qquad$ a) Is the same true for an infinite definably simple locally finite group? A group is said to be definably simple if it does not contain proper definable normal subgroups.
$\qquad$ b) The same question for an infinite definably simple locally finite group with bounded derived lengths of its solvable subgroups.

The normal covering number $\gamma(G)$ of a finite non-cyclic group $G$ is the minimum number of proper subgroups of $G$ such that $G$ is the union of their conjugates.
$\qquad$ (ae) What is the exact value of $\lim \inf_{n \text{ even}} \gamma(S_n)/n$?
$\qquad$ (ao) What is the exact value of $\lim \inf_{n \text{ odd}} \gamma(S_n)/n$?
$\qquad$ (b) What are the exact values of $\lim \inf_{n \text{ even}} \gamma(A_n)/n$ and $\lim \inf_{n \text{ odd}} \gamma(A_n)/n$?
$\qquad$ (c) What is the exact value of $\lim \sup_{n \text{ even}} \gamma(S_n)/n$? It is known that $\lim \sup_{n \text{ odd}} \gamma(S_n)/n = 1/2$.
$\qquad$ (d) What are the exact values of $\lim \sup_{n \text{ odd}} \gamma(A_n)/n$ and $\lim \sup_{n \text{ even}} \gamma(A_n)/n$?

Let $\mathfrak{R}_{p^k}$ be the variety of class 2 nilpotent groups of exponent $p^k$, where $p$ is a prime number and $k \geqslant 2$. It is true that for every $p$ and $k$ there are infinitely many subquasivarieties of $\mathfrak{R}_{p^k}$ each of which is generated by a finite group with derived subgroup of exponent $p^k$ and does not have an independent basis of quasi-identities?

A subgroup $H$ of a group $G$ is called commensurated if for all $g \in G$, the index $|H : H \cap gHg^{-1}|$ is finite. Can a non-abelian free group (or a non-elementary hyperbolic group) contain two infinite commensurated subgroups $A$, $B$ with a trivial intersection? The answer is negative if $A$ or $B$ is normal.

Let $E(l, m, n)$ be a group on generators $a, b$ with the presentation $\langle a, b \mid a^l = 1, (ab)^m = b^n \rangle$, where $l, m, n$ satisfy $1/l + 1/m + 1/n < 1$. Are there any values for $l, m, n$ such that $E(l, m, n)$ is shortlex automatic?

It is known that $E(6, 2, 6)$ is not shortlex automatic.

(G. Verret). Does there exist a finite group $G$ with two normal subgroups $K$ and $L$, each with index 12 in $G$, such that $K$ is isomorphic to $L$, but $G/K$ is isomorphic to $C_{12}$, while $G/L$ is isomorphic to $A_4$?

Let $G$ be a non-free and non-cyclic one-relator group in which every subgroup of infinite index is free. Is $G$ a surface group?

Is there a constant $\delta$ with $0 < \delta < 1$ and an integer $N$ such that whenever $A$, $B$, and $C$ are conjugacy classes in the alternating group $\text{Alt}(n)$ each of size at least $|\text{Alt}(n)|^\delta$ with $n \geqslant N$, then $ABC = \text{Alt}(n)$?

Does the group $\text{SL}_2(\mathbb{Z}[\sqrt{2}])$ admit a faithful transitive amenable action?

This is the same as asking if it admits any co-amenable subgroup except the finite index subgroups.

Let $n = k^2$. Let $\sigma \in S_n$ be the product of $k$ disjoint cycles, of lengths $1, 3, 5, \dots, 2k - 1$. Find the limiting proportion of odd entries in the $\sigma$-column of the character table of $S_n$.

For some background, see pages 1005–1006 in (D. Gluck, Proc. Amer. Math. Soc., 147 (2019), 1005–1011).

Let $\phi$ be the Euler totient function. Does there exist a constant $a > 0$ such that $|\text{Aut}(G)| \geqslant \phi(|G|)^a$
$\qquad$ a) for every finite group $G$?
$\qquad$ b) for every finite nilpotent group $G$?

If such a constant $a$ exists, then $a \leqslant \frac{40}{41}$ (J. González-Sánchez, A. Jaikin-Zapirain, Forum Math. Sigma, 3, Article ID e7, 11 p., electronic only (2015)). Also cf. 12.77.

Let $G$ be a finite group, $p$ a prime number, and let $|g^G|_p$ denote the maximum power of $p$ that divides the class size of an element $x \in G$. Suppose that there exists a $p$-element $g \in G$ such that $|g^G|_p = \max_{x \in G} |x^G|_p$. Is it true that $G$ has a normal $p$-complement?

Let $L$ be a non-abelian finite simple group, and let $H(L) = M(L).L$ be the universal perfect central extension, where $M(L)$ is the Schur multiplier of $L$. Suppose that $G$ is a finite group such that the set of class sizes of $G$ is the same as the set of class sizes of $H(L)$. Is it true that $G \cong H(L) \times A$, where $A$ is an abelian group?

Let $S$ be a non-abelian finite simple group. Is it true that for any $n \in \mathbb{N}$, if the set of class sizes of a centreless finite group $G$ is the same as the set of class sizes of the direct power $S^n$, then $G \cong S^n$?

(P. M. Neumann and R. Vaughan-Lee). Let $G$ be a perfect and centreless finite group, and let $n$ be the maximum size of a conjugate class in $G$. Is it true that $|G| \leqslant n^2$?

Let $\omega(G)$ denote the set of element orders of a finite group $G$, and $h(G)$ the number of pairwise nonisomorphic finite groups $H$ with $\omega(H) = \omega(G)$. Find $h(L)$, where
$\qquad$ a) $L$ is the symmetric group of degree 10;
$\qquad$ b) $L$ is the automorphism group of the simple sporadic Janko group $J_2$;
$\qquad$ c) $\text{PSL}(2, q) < L \leqslant \text{Aut}(\text{PSL}(2, q))$.

See the current status in (M. A. Grechkoseeva, V. D. Mazurov, W. J. Shi, A. V. Vasil’ev, N. Yang, Commun. Math. Stat., 11, no. 2 (2023), 169–194; Subsection 4.2).

(E. Rapaport Strasser). The Lovász conjecture states that a vertex-transitive connected graph is Hamiltonian. In the special case of Cayley graphs, one can ask the following. Consider a set $A$ which generates a group $G$ and is symmetric ($x \in A$ implies $x^{-1} \in A$). Is there a list $a_1, a_2, \dots, a_n$ of elements of $A$ such that $a_1, a_1 a_2, a_1 a_2 a_3, \dots, a_1 a_2 \dots a_n$ is a complete list of the elements of $G$?

(A. Bauer). Does Higman’s Embedding Theorem relativize in the following way? Is it the case that for every subset $X \subseteq \mathbb{N}$, there is a finitely generated group $G_X$ that has an $X$-computable presentation (that is, there is a finite generating set relative to which the set of relations is computably enumerable with an $X$ oracle), and such that any finitely generated group has an $X$-computable presentation if and only if it can be embedded as a finitely generated subgroup of a quotient of a free product of finitely many copies of $G_X$ by the normal closure of a finite subset?

Higman’s Embedding Theorem says that for computable $X$, one may take $G_X = \mathbb{Z}$.

Let $f$ be an inner screen of a saturated formation $\mathfrak{F}$ and suppose that a finite group $A$ acts faithfully and $f$-hypercentrally on a finite group $G$. Is it true that $G \rtimes A \in \mathfrak{F}$ for all $G \in \mathfrak{F}$ if and only if $\mathfrak{F}$ is a Fitting class?

(Well-known problem). Let $S$ be a connected orientable hyperbolic surface of finite type and complexity at least 2. Does its mapping class group $\text{MCG}(S)$ have a non-elementary hyperbolic quotient?

See also a closely related question 16.108 about braid groups.

The free Burnside group of exponent four on three generators, $B(3, 4)$, has order $2^{69}$ as shown by Bayes, Kautsky, and Wamsley (1974). Their proof is based on a theorem of Sanov (1940) which shows that $B(n, 4)$ is finite. Sanov’s proof for $B(3, 4)$ uses more than $2^{32}$ fourth powers, because the subgroup of $B(3, 4)$ generated by two of its generators and the square of the third has order $2^{32}$. It is also known that $B(3, 4)$ needs at least 105 relations to define it, as shown by Havas and Newman (1980).
$\qquad$ a) Can $B(3, 4)$ be defined with fewer than a million fourth powers?
$\qquad$ b) Can $B(3, 4)$ be defined with fewer than a thousand fourth powers?
$\qquad$ c) What is the smallest number of fourth powers which define $B(3, 4)$?

Is it true that for every finite group $G$ and every factorization $|G| = ab$ there exist subsets $A, B \subseteq G$ with $|A| = a$ and $|B| = b$ such that $G = AB$? Cf. 19.35.

Suppose that $H$ is an almost simple group of Lie type and $G$ is a finite group such that $G$ and $H$ have the same sets of degrees of irreducible complex characters. Must there exist an abelian normal subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$?

Let $F$ be a non-abelian finitely generated free group, $1 \neq w \in F$, and $n \geqslant 1$. Is the group $\langle F, t \mid t^n = w \rangle$ linear of degree 2 over a field of characteristic 0 if $w$ is not a proper power in $F$?

This question is motivated by the following well-known question: is it true that the free $\mathbb{Q}$-group $F^{\mathbb{Q}}$ is linear over a field? (See 13.39(b).)

Let $G$ be a $\kappa$-existentially closed group of cardinality $\lambda > \kappa$, where $\kappa$ is a regular cardinal. Is it true that $|\text{Aut}(G)| = 2^\lambda$?

Suppose that $G$ is a non-cyclic residually finite group in which every subgroup of finite index (including the group itself) is defined by a single defining relation, while all infinite index subgroups are free. Is it true that $G$ is either free or isomorphic to the fundamental group of a compact surface?

See 7.36 for a negative solution of a similar question without the assumption on infinite index subgroups.

(Y. Cornulier, A. Mann, A. Thom). Does there exist a finitely generated residually finite group that is not (elementary) amenable and satisfies a nontrivial group law?

Is every family of finite groups satisfying a common group law uniformly amenable?

The definition of $\text{CT}(\mathbb{Z})$ is given in 17.57. Is it true that a finitely generated subgroup of $\text{CT}(\mathbb{Z})$ either has only finitely many orbits on $\mathbb{Z}$ or there is a set of representatives for its orbits on $\mathbb{Z}$ which has positive density?

Let $n$ be a positive integer, and let $G \leqslant \text{GL}(n, \mathbb{Z})$ be finitely generated. Given a bound $b \in \mathbb{N}$, let $e_b$ be the number of elements of $G$ all of whose matrix entries have absolute value $\leqslant b$. Does the limit $\lim_{b \to \infty} \ln e_b / \ln b$ always exist?

A group action on a compact space is said to be topologically free if the set of points with trivial stabilizer is dense. Let $G$ be a locally compact group, and $\partial_{\text{sp}} G$ its Furstenberg boundary (the largest minimal and strongly proximal compact $G$-space). Let $\Gamma_1$ and $\Gamma_2$ be two lattices in $G$, both acting faithfully on $\partial_{\text{sp}} G$.
$\qquad$ a) Is it possible that the $\Gamma_1$-action on $\partial_{\text{sp}} G$ is topologically free, but the $\Gamma_2$-action on $\partial_{\text{sp}} G$ is not topologically free?
$\qquad$ b) If yes, can this also happen if $\partial_{\text{sp}} G = G/H$ is a homogeneous $G$-space?

Let $G = F_n$ be a finitely generated free group. Let $X$ be a compact $G$-space on which the $G$-action is faithful, minimal, and strongly proximal. Does it follow that the action is topologically free?

Suppose that $P$ is a finite $p$-group with a non-trivial partition (which is equivalent to having proper Hughes subgroup $H_p(P) := \langle g \in P \mid g^p \neq 1 \rangle \neq P$). If $P$ admits a fixed-point-free automorphism of prime order, must $P$ be of exponent $p$?

Is it true that any finite group contains a 2-generated subgroup with the same exponent?

For a positive integer $m$, let $G_m$ be the largest group generated by $m$ involutions such that $(xy)^4 = 1$ for any two involutions $x, y \in G_m$. What is the order of $G_4$?

It is known that the order of $G_3$ is equal to $2^{11}$. Also cf. 18.58.

Is it true that for every odd integer $t > 3$ there exists a finite non-abelian group $G$ of odd order with exactly $t$ conjugacy classes?

A famous result of Burnside states that if $k(G)$ is the number of conjugacy classes of a finite group $G$ of odd order, then $|G| - k(G)$ is divisible by 16. Is it true that for every integer $m > 0$ there exists a finite non-abelian group $G(m)$ of odd order such that $|G(m)| - k(G(m)) = 16m$?

A nonisotropic unitary graph $\Gamma$ is distance-regular with intersection array $\{q(q - 1), (q + 1)(q - 2), q + 1; 1, 1, q(q - 2)\}$ for some prime power $q$. The group $G = \text{Aut}(\Gamma)$ acts transitively on the vertex set and on the edge set of $\Gamma$. It is known that $\Gamma$ is distance-transitive if $q = 3$. Does there exist a distance-regular graph with such an intersection array if $q$ is not a prime power?

A distance-regular graph of diameter 3 with the second eigenvalue $\theta_1 = a_3$ is called a Shilla graph. For a Shilla graph $\Gamma$ the number $a = a_3$ divides $k$ and we set $b = b(\Gamma) = k/a$. Koolen and Park proved that there are 12 feasible intersection arrays of Shilla graphs with $b = 3$. At present it is proved that a Shilla graph with $b = 3$ has intersection array $\{12, 10, 3; 1, 3, 8\}$ (Doro graph), $\{12, 10, 5; 1, 1, 8\}$ (nonisotropic unitary graph for $q = 4$), or $\{15, 12, 6; 1, 2, 10\}$. The automorphisms of the last graph were found by A. Makhnev and N. Zyulyarkina (Doklady Maths., 84, no. 1 (2011), 510–514). Does the graph with intersection array $\{15, 12, 6; 1, 2, 10\}$ exist?

Are there soluble finite groups $G$ and $H$, of derived lengths 2 and 4 and having identical character tables?

Pairs of nonisomorphic soluble finite groups with identical character tables and with derived lengths $n$ and $n + 1$ for any $n \geqslant 2$ were constructed in (S. Mattarei, J. Algebra, 175 (1995) 157–178).

Is a periodic group $G$ locally finite if it is generated by involutions and the centralizer of every involution in $G$ is locally finite?

Is a periodic group $G$ locally finite if every finite subgroup of $G$ is contained in a subgroup of $G$ isomorphic to a finite alternating group?

Let $\omega(G)$ denote the set of element orders of a finite group $G$. A finite group $G$ is said to be recognizable (by spectrum) if every finite group $H$ with $\omega(H) = \omega(G)$ is isomorphic to $G$.
$\qquad$ (a) Is it true that for every $n$ there is a recognizable group that is the $n$-th direct power of a nonabelian simple group?
$\qquad$ (b) Is it true that there is a nonabelian simple group $L$ such that for every $n$ there is a recognizable group whose socle is the $k$-th direct power of $L$ for some $k \geqslant n$?

A subgroup $H$ is called a virtual retract of a group $G$ if $H$ is a retract of a finite-index subgroup of $G$. Is it true that every finitely generated subgroup of a finitely generated virtually free group is a virtual retract?

For motivation and partial results see (A. Minasyan, Int. Math. Res. Notes, 2021, no. 17 (2021), 13434–13477).

We say that $G$ is a virtually compact special group if $G$ has a finite-index subgroup which is isomorphic to the fundamental group of a compact special complex (in the sense of F. Haglund, D. T. Wise, Geom. Funct. Anal., 17, no. 5 (2008), 1551–1620). Let $G$ be a virtual retract of a finitely generated right-angled Artin group. Must $G$ be a virtually compact special group?

An affirmative answer would provide an algebraic characterization of the class of virtually compact special groups as the class groups admitting finite index subgroups that are virtual retracts of right-angled Artin groups.

Suppose that $G$ is a virtually compact special group. Is it true that the centralizer of any element in $G$ is itself virtually compact special?

(J. Dixmier) A group is said to be unitarisable if its every uniformly bounded representation on a Hilbert space is unitarisable. Is every unitarisable group amenable?

a) Prove that every unitarisable group has trivial cost.

b) At least prove that every unitarisable group has vanishing first $L^2$ Betti number.

Let $G$ be a non-amenable group.
$\qquad$ a) Prove that $G^n$ is non-unitarisable for some $n$.
$\qquad$ b) At least prove that $G^\infty$, the direct sum (restricted product), is non-unitarisable.

Given a complex irreducible character $\chi \in \text{Irr}(G)$ of a finite group $G$, let $\mathbb{Q}(\chi)$ denote the field extension of $\mathbb{Q}$ obtained by adjoining to $\mathbb{Q}$ all the values of $\chi$. We say that a finite group $G$ is $k$-rational if $|\mathbb{Q}(\chi) : \mathbb{Q}|$ divides $k$ for every $\chi \in \text{Irr}(G)$. Does there exist a real-valued function $f$ such that if $p$ is the order of a cyclic composition factor of a $k$-rational group $G$, then $p \leqslant f(k)$?

If $k = 1$, then we know that $p \leqslant 11$ by a theorem of J. G. Thompson (J. Algebra, 319 (2008), 558–594).

A Schmidt $(p, q)$-group is a finite non-nilpotent group all of whose proper subgroups are nilpotent and whose Sylow $p$-subgroup is normal. The $N$-critical graph $\Gamma_{Nc}(G)$ of a finite group $G$ is a directed graph on the vertex set of all prime divisors of $|G|$ in which $(p, q)$ is an edge of $\Gamma_{Nc}(G)$ if and only if $G$ has a Schmidt $(p, q)$-subgroup.

Suppose that a finite group $G$ is such that $G = AB = AC = BC$, where $A, B, C$ are subgroups of $G$. Is
$$\Gamma_{Nc}(G) = \Gamma_{Nc}(A) \cup \Gamma_{Nc}(B) \cup \Gamma_{Nc}(C)?$$

This is true if $A, B, C$ are soluble.

Suppose that $G$ is a finite group, $p$ is a prime, and $B$ is a Brauer $p$-block of $G$ with defect group $D$. Let $\text{cd}(B)$ be the set of degrees of the irreducible complex characters in $B$.
$\qquad$ a) Is it true that the derived length of $D$ is bounded by $|\text{cd}(B)|$?
$\qquad$ b) (A. Jaikin-Zapirain). Is is even true that $|\text{cd}(D)| \leqslant |\text{cd}(B)|$?

The submonoid membership problem for a group $G$ generated by an alphabet $A$ asks, for given words $x_1, x_2, \dots, x_n$ and a word $w$ over $A$, whether $w$ belongs to the submonoid of $G$ generated by the $x_i$. Does there exist a hyperbolic one-relator group with undecidable submonoid membership problem?

Is the submonoid membership problem decidable for every one-relator group with torsion?

As a strengthening of the Burnside restriction, for every pair $(k, n)$ of positive integers, let a group $G$ satisfy condition $C_{k,n}$ if every $k$-generated subgroup of $G$ is finite of order at most $n$.
$\qquad$ a) Does there exist $k \geqslant 2$ such that for any $n$ all groups with condition $C_{k,n}$ are locally finite?
$\qquad$ b) In particular, is it true that for any $n$ the condition $C_{2,n}$ implies local finiteness?
$\qquad$ c) Find possibly more pairs $(k, n)$ for which groups with condition $C_{k,n}$ are locally finite. (For example, all groups with condition $C_{2,20}$ are metabelian, and therefore locally finite.)

(J. Lauri). A card of a finite simple undirected graph $G$ of order $n = |V(G)|$ is an induced subgraph of order $n - 1$. Let $k$ be 2, 3, 4. For a connected graph $G$ with $k$ isomorphism types of cards, can $\text{Aut}(G)$ have more than $k$ orbits on the vertex set $V(G)$?
If a graph $G$ has $k$ isomorphism types of cards, then the group $\text{Aut}(G)$ of automorphisms of $G$ has obviously at least $k$ orbits on $V(G)$. It is known that if all cards are mutually isomorphic, then $\text{Aut}(G)$ is transitive on $V(G)$. Examples are known of graphs $G$ with 5 isomorphism types of cards for which $\text{Aut}(G)$ has 6 orbits on $V(G)$, and of graphs with 6 isomorphism types of cards for which $\text{Aut}(G)$ has 7 orbits on $V(G)$.

Is there a variety of groups $\Theta$ such that the group of automorphisms of the category of free finitely generated groups $\Theta^0$ contains outer automorphisms?

Can every countable group $G$ be factorized $G = AB$ into infinite subsets $A, B$ such that every element $g \in G$ has a unique representation $g = ab$ for $a \in A, b \in B$? This is true if $G$ is topologizable.

a) Conjecture: there are only finitely many nonabelian finite simple groups $G$ that have a normal subset $S$ closed under inversion such that $|S| > |G|/\log_2 |G|$ and $S^2 \neq G$.

(b) The same conjecture for the groups $\text{PSL}(2, p)$.

Let $G$ be a finite $p$-group and assume that all abelian normal subgroups of $G$ can be generated by $k$ elements. Is it true that every abelian subgroup of $G$ can be generated by $2k$ elements?

The $k$-th direct power of $D_{16}$ shows that this bound would be best possible.

Let $G$ be a finite $p$-group and assume that all abelian normal subgroups of $G$ have order at most $p^k$. Is it true that every abelian subgroup of $G$ has order at most $p^{2k}$?

(O. I. Tavgen’). A group is said to be boundedly generated if it is a product of finitely many cyclic groups. Is it true that every boundedly generated residually finite group is linear over a field of characteristic zero?

This is true for residually finite-soluble groups (L. Pyber, D. Segal, J. Reine Angew. Math., 612 (2007), 173–211).

For an irreducible complex character $\chi$ of a finite group $G$, the codegree of $\chi$ is defined by $\text{cod}(\chi) = |G : \text{ker}\,\chi|/\chi(1)$. Let $\text{Cod}(G)$ be the set of irreducible character codegrees of $G$.

Conjecture: If $G$ has an element of order $m$, then $m$ divides some member of $\text{Cod}(G)$.

Conjecture: Suppose that $G$ is a non-abelian simple group and $H$ is a finite group such that $\text{Cod}(G) = \text{Cod}(H)$ (see 20.78 for notation). Then $G \cong H$.

Let $G$ be a finite group, and $\chi$ an irreducible complex character of $G$. We call $\chi$ a P-character if $\chi$ is a constituent of $(1_H)^G$ for some maximal subgroup $H$ of $G$; and $\chi$ is said to be monomial if $\chi$ is induced by a linear character of a subgroup of $G$.
Conjecture: $G$ is solvable if and only if all its P-characters are monomial.

Does there exist a class $\mathfrak{X}$ of finite groups satisfying the following conditions:
$\qquad$ (1) $\mathfrak{X}$ is closed with respect to taking subgroups and homomorphic images;
$\qquad$ (2) the product of two normal $\mathfrak{X}$-subgroups of an arbitrary group is always an $\mathfrak{X}$-group;
$\qquad$ (3) for every positive integer $n$, there exist a finite group and its conjugacy class $D$ such that any $n$ elements of $D$ generate an $\mathfrak{X}$-subgroup, whereas $\langle D \rangle \notin \mathfrak{X}$.

It is known that there are no classes $\mathfrak{X}$ satisfying (1)–(3) that are closed with respect to extensions (D. O. Revin, Algebra i Analiz, 37, no. 1 (2025), 141–176 (Russian)).

Two groups are said to be isospectral if they have the same set of element orders. Suppose that $G$ is a finite group such that every finite group isospectral to $G$ is isomorphic to $G$. Is it true that the quotient of $G$ by its socle is solvable?

Let $\mathfrak{X}$ be a class of finite groups that is closed with respect to taking subgroups, homomorphic images, and extensions. A subgroup $H$ of a finite group $G$ is said to be $\mathfrak{X}$-submaximal if there exists an embedding of $G$ into a group $G^*$ such that $G$ is subnormal in $G^*$ and $H$ coincides with the intersection of $G$ and an $\mathfrak{X}$-maximal subgroup of $G^*$.

Suppose that all $\mathfrak{X}$-submaximal subgroups of a characteristic subgroup $N$ of a finite group $G$ are conjugate in $N$. Does it follow that $HN/N$ is an $\mathfrak{X}$-submaximal subgroup of $G/N$ for every $\mathfrak{X}$-submaximal subgroup $H$ of $G$?

For normal subgroups $N$, this is not true even if $N$ is an $\mathfrak{X}$-group or if $N$ does not contain nontrivial $\mathfrak{X}$-subgroups. A positive answer is known in the case where $N$ coincides with the $\mathfrak{F}$-radical of $G$ for a Fitting class $\mathfrak{F}$.

Does there exist a nilpotent group of class 3 with a non-trivial 5-th dimension subgroup?

Let $F$ be a free group. An element $\omega \in F$ is said to be primitive if there is a minimal generating system of $F$ that contains $\omega$, almost primitive if it is primitive in each finitely generated proper subgroup of $F$ containing $\omega$, tame almost primitive if, whenever $\omega^\alpha$ is contained in a subgroup $H$ of $F$ with $\alpha \geqslant 1$ minimal, either $\omega^\alpha$ is primitive in $H$ or the index of $H$ in $F$ is just $\alpha$. In (B. Fine, A. Moldenhauer, G. Rosenberger, L. Wienke, Topics in Infinite Group Theory: Nielsen Methods, Covering Spaces, and Hyperbolic Groups, De Gruyter, Berlin, 2021) it is shown that $u = [a_1, b_1][a_2, b_2] \dots [a_g, b_g]$ is tame almost primitive in the free group on $a_1, b_1, \dots, a_g, b_g$ with $g \geqslant 1$, and $v = c_1^2 \dots c_p^2$ is tame almost primitive in the free group on $c_1, \dots, c_p$ with $p \geqslant 2$.

Are there tame almost primitive elements in free groups other than $u, v$, and their product $uv$ in the free group on $a_1, b_1, \dots, a_g, b_g, c_1, \dots, c_p$?

Suppose $G$ is a finite group and $p$ a prime such that the number $s_p(G) = 1+kp$ of Sylow $p$-subgroups $P$ of $G$ is greater than 1. By a theorem of Frobenius the set $G_p$ of all $p$-elements in $G$ has cardinality $|G_p| = |P| \cdot f_p(G)$ for some positive integer $f_p(G)$, and one easily gets that $f_p(G) = 1 + \ell(p-1) \leqslant k - \frac{k-1}{p-1}$. Usually, $\ell < k$ (but $\ell = k$ if $k < p$). Is always $\ell^p \geqslant k^{p-1}$?

Recently P. Gheri showed that $f_p(G)^p \geqslant s_p(G)^{p-1}$ if $G$ is $p$-solvable (Ann. Mat. Pura Appl. (4), 200 (2021), 1231–1243).

What are the non-abelian composition factors of finite groups in which the order of every element is divisible by at most two primes?

For the case of one prime, see (https://arxiv.org/pdf/2003.09445.pdf).

Let $u$ be an element of a free group $F_r$. Is it true that there is $v \in F_r$ (that depends on $u$) that cannot be a subword of any cyclically reduced word $\phi(u)$, where $\phi$ is an automorphism of $F_r$?

An element $g$ of a group $G$ is said to be almost Engel if there is a finite subset $\mathcal{E}(g)$ of $G$ such that for every $x \in G$ all sufficiently long commutators $[\dots[[x, g], g], \dots, g]$ belong to $\mathcal{E}(g)$, that is, there is a positive integer $n(x, g)$ such that $[\dots[[x, g], g], \dots, g] \in \mathcal{E}(g)$ if $g$ is repeated $\geqslant n(x, g)$ times. An element $g$ is Engel if we can take $\mathcal{E}(g) = \{1\}$. The set of Engel elements of a linear group is a subgroup by a well-known result of Gruenberg (J. Algebra, 3 (1966), 291–303). Is the set of almost Engel elements of a linear group a subgroup?

A linear group in which all elements are almost Engel is finite-by-hypercentral (P. Shumyatsky, Monatsh. Math., 186 (2018), 711–719).

A profinite group in which all centralizers of non-trivial elements are pronilpotent is called a CN-group. Find an example of a finitely generated infinite profinite CN-group which is not prosoluble.

The structure of profinite CN-groups is described in (P. Shumyatsky, Israel J. Math., 235, no. 1 (2020), 325–347).

A group $G$ is said to be stable if for any first-order formula $\phi(\bar{x}, \bar{y})$ (in the first-order language of groups $\{\cdot, {}^{-1}, 1\}$) there exists a natural number $n$ such that whenever there exist sequences of tuples of $G$, $(a_i)_{i < m}$, $(b_i)_{i < m}$ with $G \models \phi(a_i, b_j)$ if and only if $i < j$, then $m \leqslant n$. Sela proved that all torsion-free hyperbolic groups are stable.
$\qquad$ a) Is there a non-stable hyperbolic group?
$\qquad$ b) Is there a non-stable virtually free group?
$\qquad$ c) Is $\text{SL}_2(\mathbb{Z})$ non-stable?

This question is about existence of an analogue of the Lazard correspondence for pre-Lie algebras and braces. A pre-Lie algebra $A$ is a vector space with a bilinear operation $(x, y) \to xy$ satisfying $(xy)z - x(yz) = (yx)z - y(xz)$ for every $x, y, z \in A$. A pre-Lie algebra $A$ is said to be left nilpotent if, for some $n \in \mathbb{N}$, $A \cdot (A \cdot (A \dots A)) = 0$ where $A$ appears $n$ times in the product. Recall that a set $A$ with binary operations $+$ and $\circ$ is a left brace if $(A, +)$ is an abelian group, $(A, \circ)$ is a group, and $a \circ (b+c) + a = a \circ b + a \circ c$ for every $a, b, c \in A$. Let $p$ be a prime, and $\mathbb{F}_p$ the field of $p$ elements. A left brace $A$ is called an $\mathbb{F}_p$-brace if its additive group is an $\mathbb{F}_p$-vector space such that $a \ast (\alpha b) = \alpha(a \ast b)$ for all $a, b \in A$, $\alpha \in \mathbb{F}_p$, where $a \ast b = a \circ b - a - b$. The idea of a connection between braces and pre-Lie algebras comes from a paper by W. Rump (2014).

a) Let $A$ be an $\mathbb{F}_p$-brace of cardinality $p^k$ for some $k$. Is it true that when $p$ is sufficiently large relative to $k$, the set $A$ with the same additive operation $+$ and with the operation $\cdot$ defined as $a \cdot b = -\sum_{i=0}^{p-2} \frac{1}{2^i} ((2^i a) \ast b)$ is a pre-Lie algebra?

(b) Let $k$ be a natural number, and let $p$ be a prime number such that $p > 2^k$. Is there a bijective correspondence between $\mathbb{F}_p$-braces of cardinality $p^k$ and left nilpotent pre-Lie algebras over $\mathbb{F}_p$ of cardinality $p^k$?

An affirmative answer to any of the above questions would have consequences for the theory of set-theoretic solutions of the Yang–Baxter equation and for the theory of Hopf–Galois extensions.

A group $G$ is called a Shunkov group if for any finite subgroup $H \leqslant G$ any two conjugate elements of prime order in $N_G(H)/H$ generate a finite subgroup. Are the following well-known results of the theory of finite groups true in the class of (periodic) Shunkov groups?
$\qquad$ a) The Baer–Suzuki theorem (see 11.11).
$\qquad$ b) The Burnside–Brauer–Suzuki theorem on the existence of a normal section of order 2 in a group with a non-trivial Sylow 2-subgroup containing only one involution (see 4.75).
$\qquad$ c) Glauberman’s Z$^*$-theorem (see 10.62 and 11.13).

Does there exist an infinite periodic simple group saturated (see the definition in 14.101) with finite Frobenius groups?

Is a periodic group a Frobenius group (see 6.53) if it is saturated with finite Frobenius groups, contains an involution, and does not contain non-cyclic subgroups of order 4?

Is a periodic group a Frobenius group (see 6.53) if it has a proper non-trivial normal abelian subgroup that contains the centralizer of each of its non-identity elements?

Is a 2-group locally finite if the centralizer of every involution is locally finite?

Let $G$ be a group of permutations of the set of positive integers $\mathbb{N}$ isomorphic to the additive group of rational numbers. Must there be an element $g \in G$ such that the set $\{a - a^g \mid a \in \mathbb{N}\}$ is infinite?

Conjecture: Let $a_1 G_1, \dots, a_k G_k$, $k > 1$, be finitely many pairwise disjoint left cosets in a group $G$ with $[G : G_i] < \infty$ for all $i = 1, \dots, k$. Then $\text{gcd}([G : G_i], [G : G_j]) \geqslant k$ for some $1 \leqslant i < j \leqslant k$.

Conjecture: Let $n$ be a positive integer, and let $G$ be a group containing no elements of order among $2, \dots, n+1$. Then, for any $A$ and $G$ with $|A| = n$, we may write $A = \{a_1, \dots, a_n\}$ with $a_1, a_2^2, \dots, a_n^n$ pairwise distinct.

Let $G$ be a finitely generated branch group. Are all finite-index maximal subgroups of $G$ necessarily normal? Cf. 18.81.

(Y. Barnea, A. Shalev). Let $G$ be a finitely generated pro-$p$ group. Must $G$ be $p$-adic analytic if it has finite Hausdorff spectrum with respect to
$\qquad$ a) the $p$-power series?
$\qquad$ b) the iterated $p$-power series?
$\qquad$ c) the lower $p$-series?
$\qquad$ d) the Frattini series?
$\qquad$ e) the dimension subgroup series?

Let $G_\Gamma$ be a partially commutative soluble group of derived length $n \geqslant 3$ with defining graph $\Gamma$ (the definition is similar to the case of $n = 2$, see 17.104). Is $G_\Gamma$ a torsion-free group?

(Well-known questions). Suppose that a group $G$ is finitely generated and decomposable into a direct product $G = G_1 \times G_2$.
$\qquad$ a) Is it true that the elementary theory of $G$ is decidable if and only if the elementary theories of the groups $G_1$ and $G_2$ are decidable?
$\qquad$ b) Is it true that the universal theory of $G$ is decidable if and only if the universal theories of the groups $G_1$ and $G_2$ are decidable?

Is there a perfect locally nilpotent $p$-group, for some prime $p$, whose proper subgroups are hypercentral?

Let $G$ be a residually finite 2-group and let $x \in G$ be a left 3-Engel element of order 2. Is $\langle x^G \rangle$ locally nilpotent?

It is known that in any group a 3-Engel element of odd order belongs to the locally nilpotent radical (E. Jabara, G. Traustason, Proc. Amer. Math. Soc., 147, no. 5 (2019), 1921–1927).

Comment of 2025: An element $a \in G$ is called a strong left 3-Engel element if $\langle a, a^g \rangle$ is nilpotent of class at most 2 and $\langle a, a^g, a^h \rangle$ is nilpotent of class at most 3 for all $g, h \in G$. (This is equivalent to $a$ being left 3-Engel when $a$ is of odd order.) It is proved that if $a$ is a strong left 3-Engel element in an arbitrary group $G$, then $\langle a \rangle^G$ is locally nilpotent (A. Hadjievangelou, G. Traustason, Proc. Amer. Math. Soc., 152, no. 4 (2024), 1467–1477).

Let $G$ be a group of exponent 8 and $x \in G$ a left 3-Engel element of order 2. Is $\langle x^G \rangle$ locally finite?

The holomorph $\text{Hol}(G)$ of a group $G$ can be defined as the normalizer of the subgroup of left translations in the group of all permutations of the set $G$. The multiple holomorph $\text{NHol}(G)$ of $G$ is the normalizer of the holomorph. Set $T(G) = \text{NHol}(G)/\text{Hol}(G)$.
$\qquad$ a) Is there a centerless group $G$ for which $T(G)$ is not an elementary abelian 2-group?
$\qquad$ b) Is there a finite centerless group $G$ for which $T(G)$ is not an elementary abelian 2-group?
$\qquad$ c) (A. Caranti). Is there a finite $p$-group $G$ for which the order of $T(G)$ has a prime divisor not dividing $(p-1)p$?

A skew brace is a set $B$ equipped with two operations $+$ and $\cdot$ such that $(B, +)$ is an additively written (but not necessarily abelian) group, $(B, \cdot)$ is a multiplicatively written group, and $a \cdot (b + c) = ab - a + ac$ for any $a, b, c \in B$.

Is there a finite skew brace with perfect additive group and non-perfect almost simple multiplicative group?

Are there residually finite hereditarily just infinite groups that are
$\qquad$ a) amenable but not solvable?
$\qquad$ b) amenable but not elementary amenable?
$\qquad$ c) of intermediate word growth?
$\qquad$ d) of intermediate subgroup growth?
All examples that we know are either linear (hence Tits Alternative applies), or have a quotient with property (T) (hence cannot be amenable).

Are there residually finite hereditarily just infinite groups admitting a self-similar action on a rooted tree that are not linear?

Let $\mathfrak{F}$ be a hereditary saturated formation and let $w^*\mathfrak{F}$ denote the class of all finite groups $G$ for which $\pi(G) \subseteq \pi(\mathfrak{F})$ and the normalizers of all Sylow subgroups of $G$ are $\mathfrak{F}$-subnormal in $G$. It is known that $w^*\mathfrak{F}$ is a formation. Must $w^*\mathfrak{F}$ be a saturated formation?

Let $H(q, c) = \langle a, b, c, d \mid [b, a] = [d, c]$, of exponent $q$, nilpotent of class $c \rangle$.
$\qquad$ a) Is it true that the Schur multiplier $M(H(8, 12))$ has exponent 32?
$\qquad$ b) Is it true that the Schur multiplier $M(H(7, 13))$ has exponent 49?
The difficulty is that these groups are too big to compute using current versions of the $p$-Quotient Algorithm, which use 32 bit arithmetic. So to tackle these groups it would help to have a version of the $p$-Quotient Algorithm using 64 bit arithmetic.

Is there a positive integer $c$ such that for any $q$ the exponent of the Schur multiplier of a finite group of exponent $q$ divides $q^c$?

Let $\chi$ be a complex irreducible character of a finite group $G$. If $\chi(x) \neq 0$ for some $x \in G$, must the order $o(x)$ of $x$ divide $|G|/\chi(1)$?

This is known to be true if $G$ is solvable, and it is known that $(o(x)\chi(1))^4$ divides $|G|^5$ for arbitrary $G$.

(Well-known problem). Does a profinite torsion group have finite exponent?

This problem reduces to the case of pro-$p$ groups (cf. W. Herfort, Arch. Math., 33 (1980), 404–410). Also cf. 3.41.

Let $G$ be a pro-$p$ group that is a 3-dimensional Poincaré duality group. Is $G$ coherent?

A group is said to be coherent if each of its finitely generated subgroups is finitely presented, and in the question the coherency is used in the pro-$p$ sense. The question is a famous problem for abstract groups, but has a positive answer for 3-manifold groups.

Let $G$ be a pro-$p$ group that is a 3-dimensional Poincaré duality group. Can it contain a direct product $F_2 \times F_2$ of free pro-$p$ groups of rank 2?

If it can, then the answer to 20.117 is negative, but in the abstract case it is known that it can not (P. H. Kropholler, M. A. Roller, J. London Math. Soc. (2), 39 (1989), 271–284).

(G. Wilkes). A pro-$p$ group is said to be accessible if there is a number $n = n(G)$ such that any finite proper reduced graph of pro-$p$ groups with finite edge groups having fundamental group isomorphic to $G$ has at most $n$ edges (J. Algebra, 525 (2019), 1–18). Is every finitely presented pro-$p$ group accessible?

Let $G$ be a locally nilpotent group with an automorphism (of infinite order). Can $G$ be simple as a group with automorphism?

Let $G$ be a finite almost simple group such that $\text{Soc}(G)$ is a non-soluble group of Lie type over a field of characteristic $p$. Let $R$ be a Sylow $q$-subgroup of $G$ (for some $q$) and let $\text{Min}_G(R)$ be the subgroup of $R$ generated by all minimal by inclusion intersections of the form $R \cap R^g$, where $g \in G$. Is it true that for $p > 3$ the subgroup $\text{Min}_G(R)$ is non-trivial if and only if the following hold: $G = \text{Aut}(L_2(p))$, where $p$ is a Mersenne prime, $\text{Min}_G(R) = R$, and $q = 2$.

It is known that this is not always the case for $p = 2, 3$.

For nilpotent subgroups $A, B, C$ of a finite group $G$, let $\text{Min}_G(A, B, C)$ be the subgroup of $A$ generated by all minimal by inclusion intersections of the form $A \cap B^x \cap C^y$, where $x, y \in G$, and let $\text{min}_G(A, B, C)$ be the subgroup of $\text{Min}_G(A, B, C)$ generated by all intersections of this kind of minimal order.
$\qquad$ a) Is it true that $\text{min}_G(A, B, C) \leqslant F(G)$?
$\qquad$ b) Is it true that $\text{Min}_G(A, B, C) \leqslant F(G)$?
$\qquad$ c) The same questions for soluble groups.

A finite group is called a $D_\pi$-group if any two of its maximal $\pi$-subgroups are conjugate.
$\qquad$ a) Is it true that for any finite $D_\pi$-group $G$ and a $\pi$-Hall subgroup $H$ of $G$, there are elements $x, y, z \in G$ such that $O_\pi(G) = H \cap H^x \cap H^y \cap H^z$?
$\qquad$ b) Suppose that $G$ is a finite $D_\pi$-group in which all simple non-abelian composition factors are sporadic or alternating groups, and let $H$ be a Hall $\pi$-subgroup of $G$. Is it true that $H \cap H^x \cap H^y = O_\pi(G)$ for some $x, y \in G$?
$\qquad$ c) Suppose that $G$ is a finite $D_\pi$-group with trivial soluble radical in which all simple non-abelian composition factors are sporadic groups, and let $H$ be a Hall $\pi$-subgroup of $G$. Is it true that $H \cap H^g = O_\pi(G)$ for some $g \in G$?

A Rota–Baxter operator on a group $G$ is a mapping $B : G \to G$ such that $B(g)B(h) = B(g B(g) h B(g)^{-1})$ for all $g, h \in G$. Let $F$ be a non-abelian free group. Is there a Rota–Baxter operator on $F$ such that its image is equal to the derived subgroup $[F, F]$?

Does there exist a non-abelian group $G$ and a Rota–Baxter operator $B : G \to G$ such that $B$ is surjective but not injective?

A brace $(G; +, \circ)$ is non-empty set $G$ with two binary operations $+$, $\circ$ such that $(G, +)$ is an additively written abelian group, $(G, \circ)$ is a multiplicatively written group, and $a \circ (b + c) + a = (a \circ b) + (a \circ c) for all a, b, c \in G$. Does there exist a brace with finitely generated group $(G, +)$ such that
$\qquad$ a) the group $(G, \circ)$ is non-solvable?
$\qquad$ b) the group $(G, \circ)$ contains a non-abelian free group?