Unsolved Problems

Let $n$ be a positive integer. For a finite group $K$ and an automorphism $\varphi$ of $K$ of order dividing $n$, let $$ X_{n,\varphi}(K):=\{x\in K\m...

Let $S$ be a finite simple group, and let $G$ be a finite group for which there exists a bijection $f:G\to S$ such that $|x|$ divides $|f(x)|$ for all...

Let $G=A_n$ or $S_n$ and let $H,K$ be soluble subgroups of $G$. For all sufficiently large $n$, can we always find an element $x\in G$ such that $H\ca...

Let $G$ be a finite group with trivial solvable radical and let $H_1,\ldots,H_5$ be solvable subgroups of $G$. Then do there always exist elements $x_...

Let $p$ be a prime. Let $G$ be a transitive subgroup of the group of finitary permutations $\operatorname{FSym}(\Omega)$ of a set $\Omega$, let $N$ be...

Let p be a prime. A totally imprimitive p-group H of finitary permutations is said to have the cyclic-block property if in the cycle decomposition of ...

(Well-known problem). A finite group G is called an IYB-group if it is isomorphic to the permutation group of a finite involutive non-degenerate set-t...

As in 17.57, let $r(m)=\{r+km\mid k\in\mathbb Z\}$ for integers $0\leqslant r<m$; for $r_1(m_1)\cap r_2(m_2)=\emptyset$ let the class transposition $\...

Let $F$ be a non-abelian free pro-$p$ group of finite rank. Can one find a finite collection $U_1,\ldots,U_n$ of open subgroups of $F$, including $F$ ...

We call a group presentation finite if it represents a finite group. We say that a presentation is just finite if it is finite and is no longer finite...

Can some or all groups of the following sorts be written as homomorphic images of nonprincipal ultraproducts of countable families of groups?...

Suppose that $U$ is a nonprincipal ultrafilter on $\omega$, and $B$ is a group such that every element $b\in B$ belongs to a subgroup of $B$ that is a...

Does $\mathbb Z^\omega$ have a subgroup whose dual is free abelian of still larger rank (the largest possible being $2^{2^{\aleph_0}}$)?...

Suppose $\alpha$ is an endomorphism of a group G such that for every group H and every homomorphism $f:G\to H$, there exists an endomorphism $\beta_f$...

Suppose B is a subgroup of the symmetric group $S_\Omega$ on an infinite set $\Omega$. Will the amalgamated free product $S_\Omega *_B S_\Omega$ of tw...

Let the width of a group (respectively, a monoid) H with respect to a generating set X mean the supremum over h $\in$ H of the least length of a group...

If $X$ is a class of groups, let $H(X)$ denote the class of homomorphic images of groups in $X$, let $S(X)$ denote the class of groups isomorphic to s...

Suppose that G is a finite group, and $A_1,A_2,A_3$ are subsets of G such that the multiplication map $A_1\times A_2\times A_3\to G$ is bijective. Mus...

Suppose that S and M are groups of finite Morley rank, S is an infinite group, and M is a non-trivial connected group definably and faithfully acting ...

Prove that a simple group of finite Morley rank without involutions cannot act definably, faithfully, and irreducibly on a connected group other than ...

Prove that a simple (that is, without proper non-trivial connected normal subgroups) algebraic group M over an algebraically closed field cannot act d...

Is the (standard, restricted) wreath product $G\wr H$ of two finitely generated Hopfian groups Hopfian?...

A graph is called a cograph if it has no induced subgraph isomorphic to a path with 4 vertices. A graph is said to be chordal if it has no induced cyc...

For a finite group G, the power graph P(G) is the graph with vertex set G and edges \{x, y\} for all $x\ne y\in G$ such that either $x\in\langle y\ran...

Let $G$ be a finite simple group and let $p_1,p_2$ be any (not necessarily distinct) prime divisors of $|G|$. Then can we always find Sylow $p_i$-subg...

Let $G$ be a non-trivial finite group and let $p_1,\ldots,p_k$ be the distinct prime divisors of $|G|$. For each $i$, let $H_i$ be a Sylow $p_i$-subgr...

A permutation on a set $\Omega$ is called a derangement if it has no fixed points in $\Omega$. Let G be a finite simple transitive permutation group. ...

Let $G$ be a finite simple transitive permutation group, and let $\delta(G)$ be the proportion of derangements in $G$. Is it true that $\delta(G)\geqs...

Let $G\leqslant\operatorname{Sym}(\Omega)$ be a finite primitive permutation group with a regular suborbit (that is, $G$ has a trivial 2-point stabili...

(Well-known question). A discrete group G is said to have the Haagerup property (also known as Gromov's a-T-menability property) if there exists a met...

Conjecture: If N is a finite soluble group, then any regular subgroup in the holomorph Hol(N) of N is also soluble....

Is the following problem decidable, and if so, what is its complexity? Given a finite group G, is there a finite group H such that the derived subgrou...

Does an analogue of Dunwoody's theorem hold for totally disconnected locally compact groups, that is, must a tdlc group of rational discrete cohomolog...

(Well-known problem). A group $G$ is a unique product group if, for any nonempty finite subsets $A,B$ of $G$, there exists an element of $G$ which can...

Let $G$ be a finite group, $w$ a multilinear commutator group-word, and $p$ a prime. Suppose that $p$ divides the order $|xy|$ whenever $x$ is a $w$-v...

Let a hierarchy of tdlc groups $\mathbf H\mathcal K$ be defined analogously to Kropholler's hierarchy in 15.45, with $\mathcal K$ being the class of p...

By definition, a constructible totally disconnected, locally compact (tdlc) group is the result of a sequence of profinite extensions and ascending HN...

The spread of a group $G$ is the greatest nonnegative integer $k$ such that for all nontrivial elements $x_1,\ldots,x_k\in G$ there exists $y\in G$ su...

Are there any locally finite, characteristically simple groups with finitely many orbits under automorphisms that are not residually finite?...

Let G be a subgroup of GL(n, Q) with finitely many orbits under automorphisms. Is G a virtually soluble group?...

A group is said to be self-similar if it admits a faithful state-closed representation by automorphisms of a regular one-rooted m-tree for some m. Can...

Let $\mathcal T_{d,c}$ denote the class of $d$-generated, torsion-free nilpotent groups having class $c$. Are there $\mathcal T_{3,3}$-groups that are...

Conjecture: Suppose that for a fixed positive integer $k$ at least half of the elements of a finite group $G$ have order $k$. Then $G$ is solvable....

Let $W_n=A_5\wr\cdots\wr A_5$ be the $n$-times iterated permutational wreath product of $A_5$ in its natural action (so $W_n$ acts on $5^n$ points), a...

(Well-known problem). Does there exist a finitely presented (infinite) simple group requiring more than two generators?...

(Well-known problem). Does there exist a finitely presented (infinite) simple group of finite cohomological dimension greater than 2?...

(Well-known problem). Does there exist a finitely presented group $G$ such that $G\cong G\times H$ for some non-trivial group $H$?...

A quasimorphism on a group $G$ is a function $f:G\to\mathbb R$ such that the quantity $\sup_{g,h}|f(g)+f(h)-f(gh)|$ is finite. A quasimorphism is homo...

An isometric action of a group G on a metric space S is called acylindrical if for every $\varepsilon>0$ there exist R, N > 0 such that for every two ...

Does every finite 3-group $T$ have a nontrivial characteristic subgroup $C$ such that if $T$ is a Sylow 3-subgroup of a finite group $G$, then $T\cap ...