Find an infinite finitely generated group with an identical relation of the form $x^{2^n} = 1$.

a) Find an infinite finitely generated group of exponent $< 100$.
b) Do there exist such groups of exponent 5?

Construct a finitely presented group with insoluble word problem and satisfying a non-trivial law.

Construct a finitely presented group with undecidable word problem all of whose non-trivial defining relations have the form $A^2 = 1$. This problem is interesting for topologists.

a) (J. Milnor). Is it true that an arbitrary finitely generated group has either polynomial or exponential growth?
b) Is it true that an arbitrary finitely presented group has either polynomial or exponential growth?
c) Is it true that every finitely generated group with undecidable word problem has exponential growth?

(P. Hall). Are the projective groups in the variety of metabelian groups free?

(Well-known problem). For which ring epimorphisms $R \to Q$ is the corresponding group homomorphism $SL_n(R) \to SL_n(Q)$ an epimorphism (for a fixed $n \geqslant 2$)? In particular, for what rings $R$ does the equality $SL_n(R) = E_n(R)$ hold?

Suppose $G$ is a finitely generated free-by-cyclic group. Is $G$ finitely presented?

Let $G$ be a finitely generated torsion-free nilpotent group. Are there only a finite number of non-isomorphic groups in the sequence $\alpha G, \alpha^2 G, \dots$? Here $\alpha G$ denotes the automorphism group of $G$ and $\alpha^{n+1}G = \alpha(\alpha^n G)$ for $n = 1, 2, \dots$

A group $G$ is called locally indicable if every non-trivial finitely generated subgroup of $G$ has an infinite cyclic factor group. Is every torsion-free one-relator group locally indicable?

Let $F$ be the free group of rank 2 in some variety of groups. If $F$ is not finitely presented, is the multiplicator of $F$ necessarily infinitely generated?

Let $G$ be a finite group and $A$ a group of automorphisms of $G$ stabilizing a series of subgroups beginning with $G$ and ending with its Frattini subgroup. Then is $A$ nilpotent?

Prove that every finite non-abelian $p$-group admits an automorphism of order $p$ which is not an inner one.

Let $p$ be a prime number. What are necessary and sufficient conditions for a finite group $G$ in order that the group algebra of $G$ over a field of characteristic $p$ be indecomposable as a two-sided ideal? There exist some nontrivial examples, for instance, the group algebra of the Mathieu group $M_{24}$ is indecomposable when $p$ is 2.

Suppose that $\mathfrak{K}$ is a class of groups meeting the following requirements: 1) subgroups and epimorphic images of $\mathfrak{K}$-groups are $\mathfrak{K}$-groups; 2) if the group $G = UV$ is the product of $\mathfrak{K}$-subgroups $U$ and $V$ (neither of which need be normal), then $G \in \mathfrak{K}$. If $\pi$ is a set of primes, then the class of all finite $\pi$-groups meets these requirements. Are these the only classes $\mathfrak{K}$ with these properties?

If $\mathfrak{V}$ is a variety of groups, denote by $\widetilde{\mathfrak{V}}$ the class of all finite $\mathfrak{V}$-groups. How to characterize the classes of finite groups of the form $\widetilde{\mathfrak{V}}$ for $\mathfrak{V}$ a variety?

Characterize the classes $\mathfrak{K}$ of groups, meeting the following requirements: subgroups, epimorphic images and groups of automorphisms of $\mathfrak{K}$-groups are $\mathfrak{K}$-groups, but not every countable group is a $\mathfrak{K}$-group. Note that the class of all finite groups and the class of all almost cyclic groups meet these requirements.

Denote by $\mathfrak{C}$ the class of all groups $G$ with the following property: if $U$ and $V$ are maximal locally soluble subgroups of $G$, then $U$ and $V$ are conjugate in $G$ (or at least isomorphic). It is almost obvious that a finite group $G$ belongs to $\mathfrak{C}$ if and only if $G$ is soluble. What can be said about the locally finite groups in $\mathfrak{C}$?

a) Let $F$ be a free group, and $N$ a normal subgroup of it. Is it true that the Cartesian square of $N$ is $m$-reducible to $N$ (that is, there is an algorithm that from a pair of words $w_1, w_2 \in F$ constructs a word $w \in F$ such that $w_1 \in N$ and $w_2 \in N$ if and only if $w \in N$)?
b) (Well-known problem). Do there exist finitely presented groups in which the word problem has an arbitrary pre-assigned recursively enumerable $m$-degree of unsolvability?

Let $G$ be a finite group, $p$ an odd prime number, and $P$ a Sylow $p$-subgroup of $G$. Let the order of every non-identity normal subgroup of $G$ be divisible by $p$. Suppose $P$ has an element $x$ that is conjugate to no other from $P$. Does $x$ belong to the centre of $G$? For $p = 2$, the answer is positive (G. Glauberman, J. Algebra, 4 (1966), 403–420).

(J. G. Thompson). Let $G$ be a finite group, $A$ a group of automorphisms of $G$ such that $|A|$ and $|G|$ are coprime. Does there exist an $A$-invariant soluble subgroup $H$ of $G$ such that $C_A(H) = 1$?

Let $G$ be a finite simple group, $\tau$ some element of prime order, and $\alpha$ an automorphism of $G$ whose order is coprime to $|G|$. Suppose $\alpha$ centralizes $C_G(\tau)$. Is $\alpha = 1$?

Suppose $T$ is a non-abelian Sylow 2-subgroup of a finite simple group $G$.
$\qquad$ a) Suppose $T$ has nilpotency class $n$. The best possible bound for the exponent of the center of $T$ is $2^{n-1}$. This easily implies the bound $2^{n(n-1)}$ for the exponent of $T$, however, this is almost certainly too crude. What is the best possible bound?
$\qquad$ b) Is it possible that $T$ is the direct product of two proper subgroups?
$\qquad$ c) Is $T' = \Phi(T)$?
$\qquad$ d) Find a “small number” of subgroups $T_1, \dots, T_n$ of $T$ which depend only on the isomorphism class of $T$ such that $\{N_G(T_1), \dots, N_G(T_n)\}$ together control fusion in $T$ with respect to $G$ (in the sense of Alperin).

Describe all finite simple groups $G$ which can be represented in the form $G = ABA$, where $A$ and $B$ are abelian subgroups.

For a given field $k$ of characteristic $p > 0$, characterize the locally finite groups with semisimple group algebras over $k$.

Classify the irreducible matrix groups over a finite field that are generated by reflections, that is, by matrices with Jordan form $\text{diag}(-1, 1, \dots, 1)$.

Describe the groups (finite groups, abelian groups) that are the full automorphism groups of topological groups.

Describe the lattice of quasivarieties of nilpotent groups of nilpotency class 2.

The conjugacy problem for metabelian groups.

Let $\mathfrak{K}_n$ be the class of all groups with a single defining relation in the variety of soluble groups of derived length $n$.
$\qquad$ a) Under what conditions does a $\mathfrak{K}_n$-group have non-trivial center? Can a $\mathfrak{K}_n$-group, $n \geqslant 2$, that cannot be generated by two elements have non-trivial centre?
$\qquad$ b) Describe the abelian subgroups of $\mathfrak{K}_n$-groups.
$\qquad$ c) Investigate the periodic subgroups of $\mathfrak{K}_n$-groups.

Let $v$ be a group word, and let $\mathfrak{K}_v$ be the class of groups $G$ such that there exists a positive integer $n = n(G)$ such that each element of the verbal subgroup $vG$ can be represented as a product of $n$ values of the word $v$ on the group $G$.
$\qquad$ a) For which $v$ do all finitely generated soluble groups belong to the class $\mathfrak{K}_v$?
$\qquad$ b) Does the word $v(x, y) = x^{-1}y^{-1}xy$ satisfy this condition?

Is there an infinite locally finite simple group satisfying the minimum condition for $p$-subgroups for every prime $p$?

Is there an infinite locally finite simple group $G$ with an involution $i$ such that the centralizer $C_G(i)$ is a Chernikov group?

Is there an infinite locally finite simple group $G$ that cannot be represented by matrices over a field and is such that for some prime $p$ the $p$-subgroups of $G$ are either of bounded derived length or of finite exponent?

Classify the composition factors of automorphism groups of finite (nilpotent of class 2) groups of prime exponent.

A countable group $U$ is said to be SQ-universal if every countable group is isomorphic to a subgroup of a quotient group of $U$. Let $G$ be a group that has a presentation with $r \geqslant 2$ generators and at most $r - 2$ defining relations. Is $G$ SQ-universal?

Let $C$ be a fixed non-trivial group (for instance, $C = \mathbb{Z}/2\mathbb{Z}$). As it is shown in (Yu. I. Merzlyakov, Algebra and Logic, 9, no. 5 (1970), 326–337), for any two groups $A$ and $B$, all split extensions of $B$ by $A$ can be imbedded in a certain unified way into the direct product $A \times \operatorname{Aut} (B \wr C)$. How are they situated in it?

A point $z$ in the complex plane is called free if the matrices $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ z & 1 \end{pmatrix}$ generate a free group. Are all points outside the rhombus with vertices $\pm 2$, $\pm i$ free?

For what natural numbers $n$ does the following equality hold:
$$GL_n(\mathbb{R}) = D_n(\mathbb{R}) \cdot O_n(\mathbb{R}) \cdot UT_n(\mathbb{R}) \cdot GL_n(\mathbb{Z})?$$ For notation, see, for instance, (M. I. Kargapolov, Ju. I. Merzljakov, Fundamentals of the Theory of Groups, Springer, New York, 1979). For given $n$ this equality implies the affirmative solution of Minkowski’s problem on the product of $n$ linear forms (A. M. Macbeath, Proc. Glasgow Math. Assoc., 5, no. 2 (1961), 86–89), which remains open for $n \geqslant 6$. It is known that the equality holds for $n \leqslant 3$ (Kh. N. Narzullayev, Math. Notes, 18 (1975), 713–719); on the other hand, it does not hold for all sufficiently large $n$ (N. S. Akhmedov, Zapiski Nauchn. Seminarov LOMI, 67 (1977), 86–107 (Russian)).

(Well-known problem). Describe the groups whose automorphism groups are abelian.

Let $G$ be a free product amalgamating proper subgroups $H$ and $K$ of $A$ and $B$, respectively.
$\qquad$ a) Suppose that $H, K$ are finite and $|A : H| > 2, |B : K| \geqslant 2$. Is $G$ SQ-universal (see 4.39)?
$\qquad$ b) Suppose that $A, B, H, K$ are free groups of finite ranks. Can $G$ be simple?

We call a variety of groups a limit variety if it cannot be defined by finitely many laws, while each of its proper subvarieties has a finite basis of identities. It follows from Zorn’s lemma that every variety that has no finite basis of identities contains a limit subvariety.
$\qquad$ a) Give explicitly (by means of identities or by a generating group) at least one limit variety.
$\qquad$ b) Is the set of limit varieties countable?

Does there exist a countable family of groups such that every variety is generated by a subfamily of it?

A locally finite group is said to be an A-group if all of its Sylow subgroups are abelian. Does every variety of A-groups possess a finite basis of identities?

Let $G$ and $H$ be finitely generated torsion-free nilpotent groups such that $\text{Aut}\,G \cong \text{Aut}\,H$. Does it follow that $G \cong H$?

What are the soluble varieties of groups all of whose finitely generated subgroups are residually finite?

(Well-known problem). Are knot groups residually finite?

Let $G$ be a finitely generated torsion-free group that is an extension of an abelian group by a nilpotent group. Then is $G$ almost a residually finite $p$-group for almost all primes $p$?

P. F. Pickel has proved that there are only finitely many non-isomorphic finitely generated nilpotent groups having the same family of finite homomorphic images. Can Pickel’s theorem be extended to polycyclic groups?

Are any two minimal relation-modules of a finite group isomorphic?

Let $G$ be a finite group, and $\mathbb{Z}_{(p)}$ the localization at $p$. Can every projective $\mathbb{Z}_{(p)}G$-module be uniquely, up to permutations and isomorphisms, written as a direct sum of indecomposable projective $\mathbb{Z}_{(p)}G$-modules?

It is known that the category of finitely generated projective $\mathbb{Z}_{(p)}G$-modules is not a Krull–Schmidt category (S. M. Woods, Canadian J. Math., 26, no. 1 (1974), 121–129). See also (D. Johnston, D. Rumynin, J. Algebra, to appear, https://arxiv.org/pdf/2507.21316).

Let $R$ be a commutative Noetherian ring with 1, and $\Lambda$ an $R$-algebra, which is finitely generated as $R$-module. Put
$T = \{U \in \operatorname{Mod} \Lambda \mid \exists$ an exact $\Lambda$-sequence $0 \to P \to \Lambda^{(n)} \to U \to 0$ for some $n$, with $P_{\mathfrak{m}} \cong \Lambda^{(n)}_{\mathfrak{m}}$ for every maximal ideal $\mathfrak{m}$ of $R \}$.
Denote by $G(T)$ the Grothendieck group of $T$ relative to short exact sequences.
$\qquad$ a) Describe $G(T)$, in particular, what does it mean: $[U] = [V]$ in $G(T)$?
$\qquad$ b) Conjecture: if $\dim(\max(R)) = d < \infty$, and there are two epimorphisms $\varphi : \Lambda^{(n)} \to U$, $\psi : \Lambda^{(n)} \to V$, $n > d$, and $[U] = [V]$ in $G(T)$, then $\operatorname{Ker} \varphi = \operatorname{Ker} \psi$.

Let a group $G$ be the product of two of its abelian minimax subgroups $A$ and $B$. Prove or refute the following statements:
$\qquad$ a) $A_0 B_0 \neq 1$, where $A_0 = \bigcap_{x \in G} A^x$ and similarly for $B_0$;
$\qquad$ b) the derived subgroup of $G$ is a minimax subgroup.

Let a finite group $G$ be the product of two subgroups $A$ and $B$, where $A$ is abelian and $B$ is nilpotent. Find the dependence of the derived length of $G$ on the nilpotency class of $B$ and the order of its derived subgroup.

(P. Hall). Find the smallest positive integer $n$ such that every countable group can be embedded in a simple group with $n$ generators.

(P. Hall). What is the cardinality of the set of simple groups generated by two elements, one of order 2 and the other of order 3?

Does there exist a linear function $f$ with the following property: if every abelian subgroup of a finite 2-group $G$ is generated by $n$ elements, then $G$ is generated by $f(n)$ elements?

Does there exist a finitely based variety of groups whose universal theory is undecidable?

Does there exist a non-abelian variety of groups (in particular, one that contains the variety of all abelian groups) whose elementary theory is decidable?

Does there exist a variety of groups that does not admit an independent system of defining identities?

Conjecture: $\frac{p^q - 1}{p - 1}$ never divides $\frac{q^p - 1}{q - 1}$ if $p, q$ are distinct primes. The validity of this conjecture would simplify the proof of solvability of groups of odd order (W. Feit, J. G. Thompson, Pacific J. Math., 13, no. 3 (1963), 775–1029), rendering unnecessary the detailed use of generators and relations.

Let $P$ be a presentation of a finite group $G$ on $m_p$ generators and $r_p$ relations. The deficiency $\operatorname{def}(G)$ is the maximum of $m_p - r_p$ over all presentations $P$. Let $G$ be a finite group such that $G = G' \neq 1$ and the multiplicator $M(G) = 1$. Prove that $\operatorname{def}(G^n) \to -\infty$ as $n \to \infty$, where $G^n$ is the $n$-th direct power of $G$.

Let $G$ be a finite $p$-group. Show that the rank of the multiplicator $M(G)$ of $G$ is bounded in terms of the rank of $G$.

Construct a finitely generated (infinite) characteristically simple group that is not a direct product of a simple group.

Let $G$ be a finite $p$-group, and suppose that $|G'| > p^{n(n-1)/2}$ for some non-negative integer $n$. Prove that $G$ is generated by the elements of breadths $\geqslant n$. The breadth of an element $x$ of $G$ is $b(x)$ where $|G : C_G(x)| = p^{b(x)}$.

Let $k$ be a field of characteristic different from 2, and $G_k$ the group of transformations $A = (a, \alpha) : x \mapsto ax + \alpha$, ($a, \alpha \in k$, $a \neq 0$). Extend $G_k$ to the projective plane by adjoining the symbols $(0, \alpha)$ and a line at infinity. Then the lines are just the centralizers $C_G(A)$ of elements $A \in G_k$ and their cosets. Do there exist other groups $G$ complementable to the projective plane such that the lines are just the cosets of the centralizers of elements of $G$?

Let $A$ be a group of automorphisms of a finite group $G$ which has a series of $A$-invariant subgroups $G = G_0 > \dots > G_k = 1$ such that every $|G_i : G_{i+1}|$ is prime. Prove that $A$ is supersoluble.

Is it true that every variety of groups whose free groups are residually nilpotent torsion-free, is either soluble or coincides with the variety of all groups? For an affirmative answer, it is sufficient to show that every variety of Lie algebras over the field of rational numbers which does not contain any finite-dimensional simple algebras is soluble.

(Well-known problem). Does there exist a non-abelian variety of groups
$\qquad$ a) all of whose finite groups are abelian?
$\qquad$ b) all of whose periodic groups are abelian?

a) Is every 2-group of order greater than 2 non-simple?
b) Is every binary-finite 2-group of order greater than 2 non-simple?

Let $G$ be a periodic group containing an involution $i$ and suppose that the Sylow 2-subgroups of $G$ are either locally cyclic or generalized quaternion. Does the element $iO_{2'}(G)$ of the factor-group $G/O_{2'}(G)$ always lie in its centre?

Let $G$ be a locally finite group containing an element $a$ of prime order such that the centralizer $C_G(a)$ is finite. Is $G$ almost soluble?

In 1972, A. Rudvalis discovered a new simple group $R$ of order $2^{14} \cdot 3^3 \cdot 5^3 \cdot 7 \cdot 13 \cdot 29$. He has shown that $R$ possesses an involution $i$ such that $C_R(i) = V \times F$, where $V$ is a 4-group (an elementary abelian group of order 4) and $F \cong \text{Sz}(8)$.
$\qquad$ a) Show that $R$ is the only finite simple group $G$ that possesses an involution $i$ such that $C_G(i) = V \times F$, where $V$ is a 4-group and $F \cong \text{Sz}(8)$.
$\qquad$ b) Let $G$ be a non-abelian finite simple group that possesses an involution $i$ such that $C_G(i) = V \times F$, where $V$ is an elementary abelian 2-group of order $2^n, n \geqslant 1$, and $F \cong \text{Sz}(2^m), m \geqslant 3$. Show that $n = 2$ and $m = 3$.