(P. Longobardi, M. Maj, A. H. Rhemtulla). Let $w = w(x_1, \dots, x_n)$ be a group word in $n$ variables $x_1, \dots, x_n$, and $V(w)$ the variety of groups defined by the law $w = 1$. Let $V(w^*)$ (respectively, $V(w^\#)$) be the class of all groups $G$ in which for every $n$ infinite subsets $S_1, \dots, S_n$ there exist $s_i \in S_i$ such that $w(s_1, \dots, s_n) = 1$ (respectively, $\langle s_1, \dots, s_n \rangle \in V(w)$).
$\qquad$ a) Is there some word $w$ and an infinite group $G$ such that $G \in V(w^\#)$ but $G \notin V(w)$?
$\qquad$ b) Is there some word $w$ and an infinite group $G$ such that $G \in V(w^*)$ but $G \notin V(w^\#)$?
The answer to both of these questions is likely to be "yes". It is known that in a) $w$ cannot be any of several words such as $x_1^n$, $[x_1, \dots, x_n]$, $[x_1, x_2]^2$, $(x_1x_2)^3 x_2^{-3} x_1^{-3}$, and $x_1^{a_1} \dots x_n^{a_n}$ for any non-zero integers $a_1, \dots, a_n$.

By a theorem of W. Burnside, if $\chi \in \text{Irr}(G)$ and $\chi(1) > 1$, then there exists $x \in G$ such that $\chi(x) = 0$, that is, only the linear characters are “nonvanishing”. It is interesting to consider the dual notion of nonvanishing elements of a finite group $G$, that is, the elements $x \in G$ such that $\chi(x) \neq 0$ for all $\chi \in \text{Irr}(G)$.
$\qquad$ a) It is proved (I. M. Isaacs, G. Navarro, T. R. Wolf, J. Algebra, 222, no. 2 (1999), 413–423) that if $G$ is solvable and $x \in G$ is a nonvanishing element of odd order, then $x$ belongs to the Fitting subgroup $\mathfrak{F}(G)$. Is this true for elements of even order too? (M. Miyamoto showed in 2008 that every nontrivial abelian normal subgroup of a finite group contains a nonvanishing element.)
$\qquad$ b) Which nonabelian simple groups have nonidentity nonvanishing elements? (For example, $A_7$ has.)

Let $\alpha$ and $\beta$ be faithful non-linear irreducible characters of a finite group $G$. There are non-solvable groups $G$ giving examples when the product $\alpha\beta$ is again an irreducible character (for some of such $\alpha, \beta$). One example is $G = \text{SL}_2(5)$ with two irreducible characters of degree 2. In (I. Zisser, Israel J. Math., 84, no. 1–2 (1993), 147–151) it is proved that such an example exists in an alternating group $A_n$ if and only if $n$ is a square exceeding 4. But do solvable examples exist? Evidence (but no proof) that they do not is given in (I. M. Isaacs, J. Algebra, 223, no. 2 (2000), 630–646).

Is it true that large growth implies non-amenability? More precisely, consider a number $\epsilon > 0$, an integer $k \geqslant 2$, a group $\Gamma$ generated by a set $S$ of $k$ elements, and the corresponding exponential growth rate $\omega(\Gamma, S)$ defined as in 14.7. For $\epsilon$ small enough, does the inequality $\omega(\Gamma, S) \geqslant 2k - 1 - \epsilon$ imply that $\Gamma$ is non-amenable?

(Well-known problem). Does there exist an infinite finitely generated group which is simple and amenable?

(Well-known problem). Is it true that Golod $p$-groups are non-amenable?

These infinite finitely generated torsion groups are defined in (E. S. Golod, Amer. Math. Soc. Transl. (2), 48 (1965), 103–106).

(Well-known problem). Is it true that the reduced $C^*$-algebra of a countable group without amenable normal subgroups distinct from $\{1\}$ is always a simple $C^*$-algebra with unique trace? See (M. B. Bekka, P. de la Harpe, Expos. Math., 18, no. 3 (2000), 215–230).

(S. M. Ulam). Let $G$ be the compact group $SO(3)$ of all rotations of a 3-dimensional Euclidean space, viewed as a discrete group.
$\qquad$ a) Can $G$ act non-trivially on a countable set?
$\qquad$ b) Let $G$ be any Lie group (indeed, any separable continuous group) made discrete: can $G$ act faithfully on a countable set?

An automorphism $\varphi$ of the free group $F_n$ on the free generators $x_1, x_2, \dots, x_n$ is called conjugating if $x_i^\varphi = t_i^{-1} x_{\pi(i)} t_i$, $i = 1, 2, \dots, n$, for some permutation $\pi \in S_n$ and some elements $t_i \in F_n$. The set of conjugating automorphisms fixing the product $x_1 x_2 \dots x_n$ forms the braid group $B_n$. The group $B_n$ is linear for any $n \geqslant 2$, while the group of all automorphisms $\text{Aut}\,F_n$ is not linear for $n \geqslant 3$. Is the group of all conjugating automorphisms linear for $n \geqslant 3$?

(Yu. I. Merzlyakov). Is the group of all automorphisms of the free group $F_n$ that act trivially on $F_n/[F_n, F_n]$ linear for $n \geqslant 3$?

(M. Morigi). An automorphism of a group is called a power automorphism if it leaves every subgroup invariant. Is every finite abelian $p$-group the group of all power automorphisms of some group?

Let $G$ be a group acting faithfully and level-transitively by automorphisms on a rooted tree $\mathcal{T}$. For a vertex $v$ of $\mathcal{T}$, the rigid vertex stabilizer at $v$ consists of those elements of $G$ whose support in $\mathcal{T}$ lies entirely in the subtree $\mathcal{T}_v$ rooted at $v$. For a non-negative integer $n$, the $n$-th rigid level stabilizer is the subgroup of $G$ generated by all rigid vertex stabilizers corresponding to the vertices at the level $n$ of the tree $\mathcal{T}$. The group $G$ is a branch group if all rigid level stabilizers have finite index in $G$. For motivation, examples and known results see (R. I. Grigorchuk, in: New horizons in pro-$p$ groups, Birkhäuser, Boston, 2000, 121–179).

Do there exist branch groups with Kazhdan’s $\text{T}$-property? (See 14.34.)

Do there exist finitely presented branch groups (see 15.12)?

Do there exist finitely generated branch groups (see 15.12)
$\qquad$ a) that are non-amenable?
$\qquad$ b) that are non-amenable and do not contain the free group $F_2$ on two generators?
$\qquad$ c) that contain $F_2$?
$\qquad$ d) that have exponential growth?

Is every maximal subgroup of a finitely generated branch group necessarily of finite index?

Do there exist groups whose rate of growth is $e^{\sqrt{n}}$
$\qquad$ a) in the class of finitely generated branch groups?
$\qquad$ b) in the whole class of finitely generated groups? This question is related to 9.9.

An infinite group is just infinite if all of its proper quotients are finite. Is every finitely generated just infinite group of intermediate growth necessarily a branch group?

A group is hereditarily just infinite if it is residually finite and all of its non-trivial normal subgroups are just infinite.
$\qquad$ a) Do there exist finitely generated hereditarily just infinite torsion groups?
$\qquad$ b) Is every finitely generated hereditarily just infinite group necessarily linear?
A positive answer to the question b) would imply a negative answer to a).

Let $p$ be a prime, and $\mathcal{F}_p$ the class of finitely generated groups acting faithfully on a $p$-regular rooted tree by finite automata. Any group in $\mathcal{F}_p$ is residually-$p$ (residually in the class of finite $p$-groups) and has word problem that is solvable in (at worst) exponential time. There exist therefore groups that are residually-$p$, have a solvable word problem, and do not belong to $\mathcal{F}_p$; though no concrete example is known. For instance:
$\qquad$ a) Is it true that some (or even all) the groups given in (R. I. Grigorchuk, Math. USSR–Sb., 54 (1986), 185–205) do not belong to $\mathcal{F}_p$ when the sequence $\omega$ is computable, but not periodic?
$\qquad$ b) Does $\mathbb{Z} \wr (\mathbb{Z} \wr \mathbb{Z})$ belong to $\mathcal{F}_2$? (Here the wreath products are restricted.)
See (A. M. Brunner, S. Sidki, J. Algebra, 257 (2002), 51–64) and (R. I. Grigorchuk, V. V. Nekrashevich, V. I. Sushchanskiĭ, in: Dynamical systems, automata, and infinite groups. Proc. Steklov Inst. Math., 231 (2000), 128–203).

(B. Hartley). An infinite transitive permutation group is said to be barely transitive if each of its proper subgroups has only finite orbits. Can a locally finite barely transitive group coincide with its derived subgroup?

Note that there are no simple locally finite barely transitive groups (B. Hartley, M. Kuzucuoğlu, Proc. Edinburgh Math. Soc., 40 (1997), 483–490), any locally finite barely transitive group is a $p$-group for some prime $p$, and if the stabilizer of a point in a locally finite barely transitive group $G$ is soluble of derived length $d$, then $G$ is soluble of derived length bounded by a function of $d$ (V. V. Belyaev, M. Kuzucuoğlu, Algebra and Logic, 42 (2003), 147–152).

(B. Hartley). Do there exist torsion-free barely transitive groups?

A permutation group is said to be finitary if each of its elements moves only finitely many points. Do there exist finitary barely transitive groups?

A transitive permutation group is said to be totally imprimitive if every finite set of points is contained in some finite block of the group. Do there exist totally imprimitive barely transitive groups that are not locally finite?

Suppose that a finite $p$-group $G$ has a subgroup of exponent $p$ and order $p^n$. Is it true that if $p$ is sufficiently large relative to $n$, then $G$ contains a normal subgroup of exponent $p$ and order $p^n$?

A finite group $G$ is said to be rational if every irreducible character of $G$ takes only rational values. Are the Sylow 2-subgroups of the symmetric groups $S_{2^n}$ rational?

A partition of a group is a representation of it as a set-theoretic union of some of its proper subgroups (components) that intersect pairwise trivially. Is it true that every nontrivial partition of a finite $p$-group has an abelian component?

Is it possible that $\text{Aut } G \cong \text{Aut } H$ for a finite $p$-group $G$ of order $> 2$ and a proper subgroup $H < G$?

Suppose that a finite $p$-group $G$ is the product of two subgroups: $G = AB$.
$\qquad$ a) Is the exponent of $G$ bounded in terms of the exponents of $A$ and $B$?
$\qquad$ b) Is the exponent of $G$ bounded if $A$ and $B$ are groups of exponent $p$?

(A. Mann). The dihedral group of order 8 is isomorphic to its own automorphism group. Are there other non-trivial finite $p$-groups with this property?

Is it true that every finite abelian $p$-group is isomorphic to the Schur multiplier of some nonabelian finite $p$-group?

M. R. Vaughan-Lee and J. Wiegold (Proc. R. Soc. Edinburgh Sect. A, 95 (1983), 215–221) proved that if a finite $p$-group $G$ is generated by elements of breadth $\leqslant n$ (that is, having at most $p^n$ conjugates), then $G$ is nilpotent of class $\leqslant n^2 + 1$; the bound for the class was later improved by A. Mann (J. Group Theory, 4, no. 3 (2001), 241–246) to $\leqslant n^2 - n + 1$. Is there a linear bound for the class of $G$ in terms of $n$?

Does there exist a function $f(k)$ (possibly depending also on $p$) such that if a finite $p$-group $G$ of order $p^m$ with $m \geqslant f(k)$ has an automorphism of order $p^{m-k}$, then $G$ possesses a cyclic subgroup of index $p^k$?

Suppose that all 2-generator subgroups of a finite 2-group $G$ are metacyclic. Is the derived length of $G$ bounded? This is true for finite $p$-groups if $p \neq 2$, see 1.1.8 in (M. Suzuki, Structure of a group and the structure of its lattice of subgroups, Springer, Berlin, 1956).

Is any free product of linearly ordered groups with an amalgamated subgroup right-orderable?

Let $F$ be the free group of finite rank $r$ with basis $\{x_1, \dots, x_r\}$. Is it true that there exists a number $C = C(r)$ such that any reduced word of length $n > 1$ in the $x_i$ lies outside some subgroup of $F$ of index at most $C \log n$?

For a class $\mathcal{M}$ of groups let $L(\mathcal{M})$ be the class of groups $G$ such that the normal subgroup $\langle a^G \rangle$ generated by an element $a$ belongs to $\mathcal{M}$ for any $a \in G$. Is it true that the class $L(\mathcal{M})$ is finitely axiomatizable if $\mathcal{M}$ is the quasivariety generated by a finite group?

Let $G$ be a group satisfying the minimum condition for centralizers. Suppose that $X$ is a normal subset of $G$ such that $[x, y, \dots, y] = 1$ for any $x, y \in X$ with $y$ repeated $f(x, y)$ times. Does $X$ generate a locally nilpotent (whence hypercentral) subgroup?
If the numbers $f(x, y)$ can be bounded, the answer is affirmative (F. O. Wagner, J. Algebra, 217, no. 2 (1999), 448–460).

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

Axiomatizing the basic properties of subnormal subgroups, we say that a functor $\tau$ associating with every finite group $G$ some non-empty set $\tau(G)$ of its subgroups is an ETP-functor if
$\qquad$ 1) $\tau(A)^\varphi \subseteq \tau(B)$ and $\tau(B)^{\varphi^{-1}} \subseteq \tau(A)$ for any epimorphism $\varphi : A \rightarrow B$, as well as $\{ H \cap R \mid R \in \tau(G) \} \subseteq \tau(H)$ for any subgroup $H \leqslant G$;
$\qquad$ 2) $\tau(H) \subseteq \tau(G)$ for any subgroup $H \in \tau(G)$;
$\qquad$ 3) $\tau(G)$ is a sublattice of the lattice of all subgroups of $G$.
Let $\tau$ be an ETP-functor. Does there exist a hereditary formation $\mathfrak{F}$ such that $\tau(G)$ coincides with the set of all $\mathfrak{F}$-subnormal subgroups in any finite group $G$?

Let $N$ be a nilpotent subgroup of a finite simple group $G$. Is it true that there exists a subgroup $N_1$ conjugate to $N$ such that $N \cap N_1 = 1$?

Let $R(m, p)$ denote the largest finite $m$-generator group of prime exponent $p$.
$\qquad$ a) Can the nilpotency class of $R(m, p)$ be bounded by a polynomial in $m$? (This is true for $p = 2, 3, 5, 7$.)
$\qquad$ b) Can the nilpotency class of $R(m, p)$ be bounded by a linear function in $m$? (This is true for $p = 2, 3, 5$.)
$\qquad$ c) In particular, can the nilpotency class of $R(m, 7)$ be bounded by a linear function in $m$?

My guess is “no” to the first two questions for general $p$, but “yes” to the third. By contrast, a beautiful and simple argument of Mike Newman shows that if $m \geqslant 2$ and $k \geqslant 2$ ($k \geqslant 3$ for $p = 2$), then the order of $R(m, p^k)$ is at least $p^{p^{\cdot^{\cdot^{\cdot^{p^m}}}}}$, with $p$ appearing $k$ times in the tower; see (M. Vaughan-Lee, E. I. Zelmanov, J. Austral. Math. Soc. (A), 67, no. 2 (1999), 261–271).

Is it true that the group algebra $k[F]$ of R. Thompson’s group $F$ (see 12.20) over a field $k$ satisfies the Ore condition, that is, for any $a, b \in k[F]$ there exist $u, v \in k[F]$ such that $au = bv$ and either $u$ or $v$ is nonzero? If the answer is negative, then $F$ is not amenable.

Let $G$ be a finite group of order $n$.
$\qquad$ a) Is it true that $|\text{Aut } G| \geqslant \varphi(n)$ where $\varphi$ is Euler's function?
$\qquad$ b) Is it true that $G$ is cyclic if $|\text{Aut } G| = \varphi(n)$?

a) Let $G$ be a reductive group over an algebraically closed field $K$ of arbitrary characteristic. Let $X$ be an affine $G$-variety such that, for a fixed Borel subgroup $B \leqslant G$, the coordinate algebra $K[X]$ as a $G$-module is the union of an ascending chain of submodules each of whose factors is an induced module $\text{Ind}_B^G V$ of some one-dimensional $B$-module $V$. (See S. Donkin, Rational representations of algebraic groups. Tensor products and filtration (Lect. Notes Math., 1140), Springer, Berlin, 1985). Suppose in addition that $K[X]$ is a Cohen–Macaulay ring, that is, a free module over the subalgebra generated by any homogeneous system of parameters. Is then the ring of invariants $K[X]^G$ Cohen–Macaulay?

b) Is the ring of invariants $K[M(n)^m]^{GL(n)}$ Cohen–Macaulay in all characteristics? (Here $M(n)^m$ is the direct sum of $m$ copies of the space of $n \times n$ matrices.)

We define the class of hierarchically decomposable groups in the following way. First, if $\mathfrak{X}$ is any class of groups, then let $\mathbf{H}_1 \mathfrak{X}$ denote the class of groups which admit an admissible action on a finite-dimensional contractible complex in such a way that every cell stabilizer belongs to $\mathfrak{X}$. Then the ``big'' class $\mathbf{H}\mathfrak{X}$ is defined to be the smallest $\mathbf{H}_1$-closed class containing $\mathfrak{X}$.
$\qquad$ a) Let $\mathfrak{F}$ be the class of all finite groups. Find an example of an $\mathbf{H}\mathfrak{F}$ group which is not in $\mathbf{H}_3\mathfrak{F}$ ($= \mathbf{H}_1\mathbf{H}_1\mathbf{H}_1\mathfrak{F}$).
$\qquad$ b) Prove or disprove that there is an ordinal $\alpha$ such that $\mathbf{H}_\alpha\mathfrak{F} = \mathbf{H}\mathfrak{F}$, where $\mathbf{H}_\alpha$ is the operator on classes of groups defined by transfinite induction in the obvious way starting from $\mathbf{H}_1$.

Can the question 7.28 on conditions for admissibility of an elementary carpet $\mathfrak{A} = \{\mathfrak{A}_r \mid r \in \Phi\}$ be reduced to Lie rank 1 if $K$ is a field? A carpet $\mathfrak{A}$ of type $\Phi$ of additive subgroups of $K$ is called admissible if in the Chevalley group over $K$ associated with the root system $\Phi$ the subgroup $\langle x_r(\mathfrak{A}_r) \mid r \in \Phi \rangle$ intersects $x_r(K)$ in $x_r(\mathfrak{A}_r)$. More precisely, is it true that the carpet $\mathfrak{A}$ is admissible if and only if the subcarpets $\{\mathfrak{A}_r, \mathfrak{A}_{-r}\}$, $r \in \Phi$, of rank 1 are admissible?

Let $M < G \leqslant \text{Sym}(\Omega)$, where $\Omega$ is finite, be such that $M$ is transitive on $\Omega$ and there is a $G$-invariant partition $\mathcal{P}$ of $\Omega \times \Omega \setminus \{(\alpha, \alpha) \mid \alpha \in \Omega\}$ such that $G$ is transitive on the set of parts of $\mathcal{P}$ and $M$ fixes each part of $\mathcal{P}$ setwise. (Here $\mathcal{P}$ can be identified with a decomposition of the complete directed graph with vertex set $\Omega$ into edge-disjoint isomorphic directed graphs.) If $G$ induces a cyclic permutation group on $\mathcal{P}$, then we showed (Trans. Amer. Math. Soc., 355, no. 2 (2003), 637–653) that the numbers $n = |\Omega|$ and $k = |\mathcal{P}|$ are such that the $r$-part $n_r$ of $n$ satisfies $n_r \equiv 1 \pmod k$ for each prime $r$. Are there examples with $G$ inducing a non-cyclic permutation group on $\mathcal{P}$ for any $n, k$ not satisfying this congruence condition?

Let $G$ be any non-trivial finite group, and let $X$ be any generating set for $G$. Is it true that every element of $G$ can be obtained from $X$ using fewer than $2 \log_2 |G|$ multiplications? (When counting the number of multiplications on a path from the generators to a given element, at each step one can use the elements obtained at previous steps.)

A group $G$ is a unique product group if, for any finite nonempty subsets $X, Y$ of $G$, there is an element of $G$ which can be written in exactly one way in the form $xy$ with $x \in X$ and $y \in Y$. Does there exist a unique product group which is not left-orderable?

Let $G$ be a group of automorphisms of an abelian group of prime exponent. Suppose that there exists $x \in G$ such that $x$ is regular of order 3 and the order of $[x, g]$ is finite for every $g \in G$. Is it true that $\langle x^G \rangle$ is locally finite?

Suppose that $G$ is a periodic group satisfying the identity $[x, y]^5 = 1$. Is then the derived subgroup $[G, G]$ a 5-group?

By a famous theorem of Wielandt the sequence $G_0 = G, G_1, \dots$, where $G_{i+1} = \text{Aut}\,G_i$, stabilizes for any finite group $G$ with trivial centre. Does there exist a function $f$ of natural argument such that $|G_i| \leqslant f(|G|)$ for all $i = 0, 1, \dots$ for an arbitrary finite group $G$ and the same kind of sequence?

Let $S$ be the set of all prime numbers $p$ for which there exists a finite simple group $G$ and an absolutely irreducible $G$-module $V$ over a field of characteristic $p$ such that the order of any element in the natural semidirect product $VG$ coincides with the order of some element in $G$. Is $S$ finite or infinite?

Suppose that $G$ is a periodic group containing an involution $i$ such that the centralizer $C_G(i)$ is a locally cyclic 2-group. Does the set of all elements of odd order in $G$ that are inverted by $i$ form a subgroup?

a) The Monster, $M$, is a 6-transposition group. Pairs of Fischer transpositions generate 9 $M$-classes of dihedral groups. The order of the product of a pair is the coefficient of the highest root of affine type $E_8$. Similar properties hold for Baby $B$, and $F'_{24}$ with respect to $E_7$ and $E_6$ when the product is read modulo centres ($2.B, 3.F'_{24}$). Explain this. Editors’ Comment of 2005: Some progress was made in (C. H. Lam, H. Yamada, H. Yamauchi, Trans. Amer. Math. Soc., 355, no. 9 (2007), 4107–4123).
b) Note that the Schur multiplier of the sporadic simple groups $M, B, F'_{24}$ is the fundamental group of type $E_8, E_7, E_6$, respectively. Why?

See (J. McKay, in: Finite groups, Santa Cruz Conf. 1979 (Proc. Symp. Pure Math., 37), Amer. Math. Soc., Providence, RI, 1980, 183–186) and (R. E. Borcherds, Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. I (1998), 607–616).

Is there a spin manifold such that the Monster acts on its loop space? perhaps 24-dimensional? hyper-Kähler, non-compact? See (F. Hirzebruch, T. Berger, R. Jung, Manifolds and modular forms, Vieweg, Braunschweig, 1992).

Suppose that $H$ is a subgroup of $\text{SL}_2(\mathbb{Q})$ that is dense in the Zariski topology and has no nontrivial finite quotients. Is then $H = \text{SL}_2(\mathbb{Q})$?

Suppose that a free profinite product $G \ast H$ is a free profinite group of finite rank. Must $G$ and $H$ be free profinite groups?
By (J. Neukirch, Arch. Math., 22, no. 4 (1971), 337–357) this may not be true if the rank of $G \ast H$ is infinite.

Does there exist a profinite group $G$ that is not free but can be represented as a projective limit $G = \varprojlim(G/N_\alpha)$, where all the $G/N_\alpha$ are free profinite groups of finite ranks?

The finiteness condition on the ranks of the $G/N_\alpha$ is essential. Such a group $G$ cannot satisfy the first axiom of countability (O. V. Mel’nikov, Dokl. AN BSSR, 24, no. 11 (1980), 968–970 (Russian)).

Is it true that any finitely generated $p'$-isolated subgroup of a free group is separable in the class of finite $p$-groups?

Is it true that $l_n^\pi(G) \leqslant n(G_\pi) - 1 + \max_{p \in \pi} l_p(G)$ for any $\pi$-soluble group $G$? Here $n(G_\pi)$ is the nilpotent length of a Hall $\pi$-subgroup $G_\pi$ of the group $G$ and $l_n^\pi(G)$ is the nilpotent $\pi$-length of $G$, that is, the minimum number of $\pi$-factors in those normal series of $G$ whose factors are either $\pi'$-groups, or nilpotent $\pi$-groups. The answer is known to be affirmative in the case when all proper subgroups of $G_\pi$ are supersoluble.

Given an ordinary irreducible character $\chi$ of a finite group $G$ write $p^{e_p(\chi)}$ to denote the $p$-part of $\chi(1)$ and put
$$e_p(G) = \max \{ e_p(\chi) \mid \chi \in \text{Irr}(G) \}.$$ Suppose that $P$ is a Sylow $p$-subgroup of a group $G$. Is it true that $e_p(P)$ is bounded above by a function of $e_p(G)$?

Let $F_n$ be the free group of finite rank $n$ on the free generators $x_1, \dots, x_n$. An element $u \in F_n$ is called positive if $u$ belongs to the semigroup generated by the $x_i$. An element $u \in F_n$ is called potentially positive if $\alpha(u)$ is positive for some automorphism $\alpha$ of $F_n$.
$\qquad$ a) Is the property of an element to be potentially positive algorithmically recognizable?
$\qquad$ b) Finally, $u \in F_n$ is called stably potentially positive if it is potentially positive as an element of $F_m$ for some $m \geqslant n$. Are there stably potentially positive elements that are not potentially positive?

For finite groups $G, X$ define $r(G; X)$ to be the number of inequivalent actions of $G$ on $X$, that is, the number of equivalence classes of homomorphisms $G \to \text{Aut}\,X$, where equivalence is defined by conjugation by an element of $\text{Aut}\,X$. Now define $r_G(n) := \max\{r(G; X) \mid |X| = n\}$.
Is it true that $r_G(n)$ may be bounded as a function of $\lambda(n)$, the total number (counting multiplicities) of prime factors of $n$?

A square matrix is said to be separable if its minimal polynomial has no repeated roots, and cyclic if its minimal and characteristic polynomials are equal. For a matrix group $G$ over a finite field define $s(G)$ and $c(G)$ to be the proportion of separable and of cyclic elements respectively in $G$. For a classical group $X(d, q)$ of dimension $d$ defined over the field with $q$ elements let $S(X; q) := \lim_{d \to \infty} s(X(d, q))$ and $C(X; q) := \lim_{d \to \infty} c(X(d, q))$. Independently G. E. Wall (Bull. Austral. Math. Soc., 60, no. 2 (1999), 253–284) and J. Fulman (J. Group Theory, 2, no. 3 (1999), 251–289) have evaluated $S(\text{GL}; q)$ and $C(\text{GL}; q)$, and have found them to be rational functions of $q$. Are $S(X; q)$ and $C(X; q)$ rational functions of $q$ also for the unitary, symplectic, and orthogonal groups?

For a class $\mathfrak{X}$ of groups let $g_\mathfrak{X}(n)$ be the number of groups of order $n$ in the class $\mathfrak{X}$ (up to isomorphism). Many years ago I formulated the following problem: find good upper bounds for the quotient $g_\mathfrak{V}(n)/g_\mathfrak{U}(n)$, where $\mathfrak{V}$ is a variety that is defined by its finite groups and $\mathfrak{U}$ is a subvariety of $\mathfrak{V}$. (This quotient is not defined for all $n$ but only for those for which there are groups of order $n$ in $\mathfrak{U}$.) Some progress has been made by G. Venkataraman (Quart. J. Math. Oxford (2), 48, no. 189 (1997), 107–125) when $\mathfrak{V}$ is a variety generated by finite groups all of whose Sylow subgroups are abelian. Conjecture: if $\mathfrak{V}$ is a locally finite variety of $p$-groups and $\mathfrak{U}$ is a non-abelian subvariety of $\mathfrak{V}$, then $g_\mathfrak{V}(p^m)/g_\mathfrak{U}(p^m) < p^{O(m^2)}$.

Moreover, this seems a possible way to attack the Sims Conjecture that when we write the number of groups of order $p^m$ as $p^{\frac{2}{27} m^3 + \varepsilon(m)}$ the error term $\varepsilon(m)$ is $O(m^2)$.

Which adjoint Chevalley groups (of normal type) over the integers are generated by three involutions two of which commute?

Does there exist an infinite finitely generated 2-group (of finite exponent) all of whose proper subgroups are locally finite?

Is it true that every hyperbolic group has a free normal subgroup with the factor-group of finite exponent?

Do there exist groups of arbitrarily large cardinality that satisfy the minimum condition for subgroups?

(B. Huppert). Let $G$ be a finite solvable group, and let $\rho(G)$ denote the set of prime divisors of the degrees of irreducible characters of $G$. Is it true that there always exists an irreducible character of $G$ whose degree is divisible by at least $|\rho(G)|/2$ different primes?

For a fixed prime $p$ does there exist a sequence of groups $P_n$ of order $p^n$ such that the number of conjugacy classes $k(P_n)$ satisfies $\lim_{n \to \infty} \log k(P_n)/\sqrt{n} = 0$?
Note that J. M. Riedl (J. Algebra, 218 (1999), 190–215) constructed $p$-groups for which the above limit is $2 \log p$.

Is it true that for every finite lattice $L$ there exists a finite group $G$ and a subgroup $H \leqslant G$ such that the interval $\text{Int}(H; G)$ in the subgroup lattice of $G$ is isomorphic to $L$? (Probably not.)

For every prime $p$ find a finite $p$-group of nilpotence class $p$ such that its lattice of normal subgroups cannot be embedded into the subgroup lattice of any abelian group. (Solved for $p = 2, 3$.)

a) Does there exist a sequence of identities in two variables $u_1 = 1$, $u_2 = 1, \dots$ with the following properties: 1) each of these identities implies the next one, and 2) an arbitrary finite group is soluble if and only if it satisfies one of the identities $u_n = 1$?
b) Consider the sequence $u_1 = [x, y], \dots, u_{n+1} = [[u_n, x], [u_n, y]]$. Is it true that an arbitrary finite group is soluble if and only if it satisfies one of these identities $u_n = 1$?

If $\Theta$ is a variety of groups, then let $\Theta^0$ denote the category of all free groups of finite rank in $\Theta$. It is proved (G. Mashevitzky, B. Plotkin, E. Plotkin J. Algebra, 282 (2004), 490–512) that if $\Theta$ is the variety of all groups, then every automorphism of the category $\Theta^0$ is an inner one. The same is true if $\Theta$ is the variety of all abelian groups.
$\qquad$ a) Is this true for the variety of nilpotent groups of class 2?
$\qquad$ b) The same is true if $\Theta$ is the variety of all nilpotent groups of class $\leqslant d$, for $d \geqslant 2$ (A. Tsurkov, Int. J. Algebra Comput., 17 (2007), 1273–1281). Is this true for the varieties of solvable groups? of metabelian groups?

An automorphism $\varphi$ of a category is called inner if it is isomorphic to the identity automorphism. Let $s : 1 \to \varphi$ be a function defining this isomorphism. Then for every object $A$ we have an isomorphism $s_A : A \to \varphi(A)$ and for any morphism of objects $\mu : A \to B$ we have $\varphi(\mu) = s_B \mu s_A^{-1}$.

Elements $a, b$ of a group $G$ are said to be symmetric with respect to an element $g \in G$ if $a = g b^{-1} g$. Let $G$ be an infinite abelian group, $\alpha$ a cardinal, $\alpha < |G|$. Is it true that for any $n$-colouring $\chi : G \to \{0, 1, \dots, n-1\}$ there exists a monochrome subset of cardinality $\alpha$ that is symmetric with respect to some element of $G$? This is known to be true for $n \leqslant 3$.

(R. I. Grigorchuk). Is it true that for any $n$-colouring of a free group of any rank there exists a monochrome subset that is symmetric with respect to some element of the group? This is true for $n = 2$.

Does there exist a Hausdorff group topology on $\mathbb{Z}$ such that the sequence $\{2^n + 3^n\}$ converges to zero?

A sequence $\{F_n\}$ of pairwise disjoint finite subsets of a topological group is called expansive if for every open subset $U$ there is a number $m$ such that $F_n \cap U \neq \varnothing$ for all $n > m$. Suppose that a group $G$ can be partitioned into countably many dense subsets. Is it true that in $G$ there exists an expansive sequence?

Let $G$ be a finite non-supersoluble group. Is it true that $G$ has a non-cyclic Sylow subgroup $P$ such that some maximal subgroup of $P$ has no proper complement in $G$?

Suppose that a periodic group $G$ contains a strongly isolated 2-subgroup $U$. Is it true that either $G$ is locally finite, or $U$ is a normal subgroup of $G$?

(Yu. I. Merzlyakov). Does there exist a rational number $\alpha$ such that $|\alpha| < 2$ and the matrices $\begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ \alpha & 1 \end{pmatrix}$ generate a free group?

Yu. I. Merzlyakov (Sov. Math. Dokl., 19 (1978), 64–68) proved that if the complex numbers $\alpha, \beta, \gamma$ are each at least 3 in absolute value, then the matrices $\begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix}$, $\begin{pmatrix} 1 & 0 \\ \beta & 1 \end{pmatrix}$, and $\begin{pmatrix} 1-\gamma & -\gamma \\ \gamma & 1+\gamma \end{pmatrix}$ generate a free group of rank three. Are there rational numbers $\alpha, \beta, \gamma$, each less than 3 in absolute value, with the same property?

A torsion-free group all of whose subgroups are subnormal is nilpotent (H. Smith, Arch. Math., 76, no. 1 (2001), 1–6). Is a torsion-free group with the normalizer condition
$\qquad$ a) hyperabelian?
$\qquad$ b) hypercentral?

A group $G$ is called discriminating if for any finite set of nontrivial elements of the direct square $G \times G$ there is a homomorphism $G \times G \to G$ which does not annihilate any of them (G. Baumslag, A. G. Myasnikov, V. N. Remeslennikov). A group $G$ is called squarelike if $G$ is universally equivalent (in the sense of first order logic) to a discriminating group (B. Fine, A. M. Gaglione, A. G. Myasnikov, D. Spellman). Must every squarelike group be elementarily equivalent to a discriminating group?

Suppose that a 2-group $G$ admits a regular periodic locally cyclic group of automorphisms that transitively permutes the set of involutions of $G$. Is $G$ locally finite?

Let $\mathfrak{A}$ and $\mathfrak{N}_c$ denote the varieties of abelian groups and nilpotent groups of class $\leqslant c$, respectively. Let $F_r = F_r(\mathfrak{A}\mathfrak{N}_c)$ be a free group of rank $r$ in $\mathfrak{A}\mathfrak{N}_c$. An element of $F_r$ is called primitive if it can be included in a basis of $F_r$. Does there exist an algorithm recognizing primitive elements in $F_r$?

Let $\Gamma$ be an infinite undirected connected vertex-symmetric graph of finite valency without loops or multiple edges. Is it true that every complex number is an eigenvalue of the adjacency matrix of $\Gamma$ under its natural action as a linear operator on the complex vector space of all complex-valued functions on the vertex set of $\Gamma$?

Let $\Gamma$ be an infinite directed graph, and $\overline{\Gamma}$ the underlying undirected graph. Suppose that the graph $\overline{\Gamma}$ admits a vertex-transitive group of automorphisms, and the graph $\overline{\Gamma}$ is connected and of finite valency. Does there exist a positive integer $k$ (possibly depending on $\Gamma$) such that for any positive integer $n$ there is a directed path of length at most $k \cdot n$ in the graph $\Gamma$ whose initial and terminal vertices are at distance at least $n$ in the graph $\overline{\Gamma}$?

Is it true that any irreducible faithful representation of a linear group $G$ of finite rank over a finitely generated field of characteristic zero is induced from an irreducible representation of a finitely generated dense subgroup of the group $G$?

A group $\langle x, y \mid x^l = y^m = (xy)^n = 1 \rangle$ is called the triangle group with parameters $(l, m, n)$. It is proved in (A. M. Brunner, R. G. Burns, J. Wiegold, Math. Scientist, 4 (1979), 93–98) that the triangle group $(2, 3, 30)$ has uncountably many non-isomorphic homomorphic images that are residually finite alternating groups. Is the same true for the triangle group $(2, 3, n)$ for all $n > 6$?

Let $G$ be a pro-$p$ group, $p > 2$, and $\varphi$ an automorphism of $G$ of order 2. Suppose that the centralizer $C_G(\varphi)$ is abelian. Is it true that $G$ satisfies a pro-$p$ identity?

An affirmative answer would generalize a result of A. N. Zubkov (Siberian Math. J., 28, no. 5 (1987), 742–747) saying that non-abelian free pro-$p$ groups cannot be represented by $2 \times 2$ matrices.

Define the weight of a group $G$ to be the minimum number of generators of $G$ as a normal subgroup of itself. Let $G = G_1 \ast \dots \ast G_n$ be a free product of $n$ nontrivial groups.
$\qquad$ a) Is it true that the weight of $G$ is at least $n/2$?$\qquad$ b) Is it true if the $G_i$ are cyclic?
$\qquad$ c) Is it true for $n = 3$ (without assuming the $G_i$ cyclic)?

Problem 5.53 (now answered in the affirmative) is the special case with $n = 3$ and all the $G_i$ cyclic.

(A. Mann, Ch. Praeger). Suppose that all fixed-point-free elements of a transitive permutation group $G$ have prime order $p$. If $G$ is a finite $p$-group, must $G$ have exponent $p$?

An automorphism $\varphi$ of a group $G$ is called a splitting automorphism of order $n$ if $\varphi^n = 1$ and $x x^\varphi x^{\varphi^2} \dots x^{\varphi^{n-1}} = 1$ for any $x \in G$.
$\qquad$ a) Is it true that the derived length of a $d$-generated nilpotent $p$-group admitting a splitting automorphism of order $p^n$ is bounded by a function of $d$, $p$, and $n$? This is true for $n = 1$, see 7.53.
$\qquad$ b) The same question for $p^n = 4$.

Let $p$ be a prime. A group $G$ satisfies the $p$-minimal condition if there are no infinite descending chains $G_1 > G_2 > \dots$ of subgroups of $G$ such that each difference $G_i \setminus G_{i+1}$ contains a $p$-element (S. N. Chernikov). Suppose that a locally finite group $G$ satisfies the $p$-minimal condition and has a subnormal series each of whose factors is finite or a $p'$-group. Is it true that all $p$-elements of $G$ generate a Chernikov subgroup?

Let $\mathfrak{F}$ be a saturated formation, and $G$ a finite soluble minimal non-$\mathfrak{F}$-group such that $G^\mathfrak{F}$ is a Sylow $p$-subgroup of $G$. Is it true that $G^\mathfrak{F}$ is isomorphic to a factor group of the Sylow $p$-subgroup of $\text{PSU}(3, p^{2n})$? This is true for the formation of finite nilpotent groups (V. D. Mazurov, S. A. Syskin, Math. Notes, 14, no. 2 (1973), 683–686; A. Kh. Zhurtov, S. A. Syskin, Siberian Math. J., 26, no. 2 (1987), 235–239).

Let $f(n)$ be the number of isomorphism classes of finite groups of order $n$. Is it true that the equation $f(n) = k$ has a solution for any positive integer $k$?

Is a periodic group locally finite if it has a non-cyclic subgroup of order 4 that coincides with its centralizer?

Is a periodic group locally finite if it has an involution whose centralizer is a locally finite group with Sylow 2-subgroup of order 2?

(W. Magnus). An element $r$ of a free group $F_n$ is called a normal root of an element $u \in F_n$ if $u$ belongs to the normal closure of $r$ in $F_n$. Can an element $u$ that lies outside the commutator subgroup $[F_n, F_n]$ have infinitely many non-conjugate normal roots?

(Well-known problem). Is the group $\text{Out}(F_3)$ of outer automorphisms of a free group of rank 3 linear?

Let $n$ be a positive integer and let $w$ be a group word in the variables $x_1, x_2, \dots$. Suppose that a residually finite group $G$ satisfies the identity $w^n = 1$. Does it follow that the verbal subgroup $w(G)$ is locally finite?