Unsolved Problems

In the notation of 21.52, let $\operatorname{Aut}_t(\Gamma)$ be the set of permutations $\tau\in S_D$ such that $(a,b)\sim(a^\tau,b^\tau)$ whenever $|...

Let $G$ be a finite soluble group with triality, which means that $G$ admits a group of automorphisms $S$ isomorphic to the symmetric group of degree ...

Let $q$ be a power of a prime $p$, and let $m_n(q)$ be the maximum $p$-length of $p$-solvable subgroups of $\operatorname{GL}(n,q)$. Is it true that $...

Let X be a non-empty class of finite groups of odd order closed under taking subgroups, homomorphic images, and extensions. Let H be an X-maximal subg...

We say that a product $XY=\{xy\mid x\in X,\ y\in Y\}$ of two subsets $X,Y$ of a group $G$ is direct if for every $z\in XY$ there are unique $x\in X$, ...

For a finite group $G$, let $\chi_1(G)$ denote the totality of the degrees of all irreducible complex characters of $G$ with allowance for their multi...

Let $G$ be a finite group, $\mathbb Z_{(p)}$ the localization at $p$, and $\mathbb F_p$ the field of $p$ elements. Let $\mathcal X$ be the class of $\...

For a fixed (finitely generated free)-by-cyclic group $G=F_n\rtimes\mathbb Z$, is there an algorithm that, given a finite subset $S$ of $G$, finds a f...

Is the uniform subgroup membership problem decidable for (finitely generated free)-by-cyclic groups? That is, for a fixed group $G=F_n\rtimes\mathbb Z...

Let $F$ be a field of characteristic $p>0$, and let $\Gamma$ be the principal congruence subgroup of $\operatorname{Aut}(F[x_1,\ldots,x_n])$ consistin...

Is it true that if a normal subgroup $A$ of a Sylow $p$-subgroup of a $p$-soluble finite group $G$ has exponent $p^e$, then the normal closure of $A$ ...

Suppose that $\phi$ is an automorphism of a finite soluble group $G$. Must $G$ contain a subgroup of index bounded in terms of $|\phi|$ and $|C_G(\phi...

Suppose that A is a nilpotent group of automorphisms of a finite soluble group G. Is the Fitting height of G bounded in terms of |A| and |CG(A)|? Isa...

Suppose that $\phi$ is an automorphism of a finite soluble group $G$, and let $r$ be the (Pr\"ufer) rank of the fixed-point subgroup $C_G(\phi)$. Is t...

Is there an algorithm deciding if a given one-relator group is hyperbolic?...

A group $G$ is called an orientable Poincar\'e duality group of dimension $n$ over a ring $R$ if it is of type $FP$ over $R$ and $H^i(G;RG)=0$ for $i\...

For a ring $R$, we say that a group $G$ is of type $FL(R)$ if the trivial $RG$-module $R$ admits a finite resolution by finitely generated free module...

We say that a group is a Tarski monster if it is finitely generated, not cyclic, and all of its proper non-trivial subgroups are isomorphic to each ot...

Is the conjugacy problem in $\operatorname{CT}(\mathbb Z)$ algorithmically decidable?...

Is it algorithmically decidable whether a given element $g\in\operatorname{CT}(\mathbb Z)$ (a) permutes a nontrivial partition of $\mathbb Z$ into re...

Given two distinct sets $P_1$ and $P_2$ of odd primes none of which is a subset of the other, is it true that $\langle \operatorname{CT}_{P_1}(\mathbb...

Let $\sigma=(\sigma_{ij})$, $1\leqslant i\ne j\leqslant n$, be an irreducible elementary net (carpet) of order $n\geqslant 3$ over a field $K$ (see 19...

Let $d$ be an integer that is not divisible by $n$-th powers of primes, let $x^n-d$ be an irreducible polynomial over $\mathbb Q$, let $\theta=\sqrt[n...

Let $p$ be a prime and let $G$ be a pro-$p$ group. Suppose that all of the (continuous Galois) cohomology groups $H^n(G,\mathbb F_p)$ of $G$ with coef...

Let $G$ be a finitely generated group with a fixed finite generating set $S$ and the corresponding word metric $L_S(*)$. An element $g$ is said to be ...

Do there exist finitely generated left-orderable groups with only one nontrivial conjugacy class?...

For $\sigma\in S_n$ and $\tau\in S_m$, where $n\leqslant m$, let $d_n^{\mathrm{flex}}(\sigma,\tau)=(1/n)\cdot(|\{x\in\{1,\ldots,n\}\mid \sigma(x)\ne\t...

Is a flexibly permutation-stable group always permutation-stable?...

Assume that a finite group G has a family of d-generator subgroups whose indices have no common divisor. Is it true that G can be generated by d+1 ele...

Is there a finite non-abelian group $G$ of odd order, with $k(G)$ conjugacy classes, such that $k(G)/|G|=1/17$?...

For $n>39$, is it true that the number of conjugacy classes in the symmetric group $S_n$ of degree $n$ is never a divisor of the order of $S_n$? In ot...

Let $\Gamma$ be a graph of diameter $d$. For $i\in\{1,2,\ldots,d\}$, let $\Gamma_i$ be the graph on the same vertex set as $\Gamma$ with vertices $u,w...

Let $G$ be a group and let $k\geqslant 2$. Let $H_1,\ldots,H_k$ be subgroups of $G$, and $g_1,\ldots,g_k$ elements of $G$ such that the cosets $g_1H_1...

The Gruenberg--Kegel graph (or the prime graph) GK(G) of a finite group G is a labelled graph with vertex set consisting of all prime divisors of the ...

Is there an almost simple but not simple group which is recognizable by the isomorphism type of its Gruenberg--Kegel graph?...

Is it true that a periodic group containing an involution is locally finite if the centralizer of every element of even order is locally finite?...

Is it true that for every positive rational number $r$ there exists a finite group $G$ such that $|\operatorname{Aut}(G)|/|G|=r$?...

Let w be a multilinear commutator word, and assume that G is a group where the set of w-values is covered by finitely many cyclic subgroups. Is it tru...

Suppose that $A$ and $G$ are finite groups such that $A$ acts coprimely on $G$ by automorphisms. Let $C=C_G(A)$ be the fixed-point subgroup, and let $...

Which finite almost simple groups are the automorphism groups of regular polytopes of rank 3? In other words, which finite almost simple groups are ge...

Let $V$ be a variety generated by a finite group, and let $f(n)$ be the order of the free group in $V$ on $n$ generators. Is it true that the sequence...

A Hausdorff topological group G is called minimal if it does not admit a strictly coarser Hausdorff group topology. A topological group is called Raik...

A group word $w$ is said to be concise in a class $\mathcal C$ of groups if for every group $G$ in $\mathcal C$ such that the set $G_w$ of word values...

A first order formula $\phi(x)$ in the group language with one free variable is said to be concise in a class $\mathcal C$ of groups if for every grou...

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 $m$...

For a finite group $G$ let $\operatorname{Cod}(G)$ denote the set of irreducible character codegrees of $G$ (see 20.78). Define $\sigma(G)=\max\{|\pi(...

Let $S$ be a finite simple nonabelian group that is not isomorphic to any group ${}^2B_2(q)$. A nonidentity automorphism $x$ of $S$ is called a $\tau$...

A nonempty class $\mathcal X$ of finite groups is said to be complete if $\mathcal X$ is closed under taking subgroups, homomorphic images, and extens...

Let $G$ be a finite group and $p$ be a prime. Let $\Psi_{p,G}$ be the class function of $G$ which vanishes on all $p$-singular elements of $G$ and who...

A finite group G is called weakly ab-maximal if |H : [H, H]| $\leqslant$ |G : [G, G]| for all H $\leqslant$ G. Do weakly ab-maximal groups have bounde...