Unsolved Problems

Let $p$ be a prime, and $P$ a finite $p$-group. (a) Suppose that $P$ has an abelian subgroup of order $p^n$. For which $n$ does $P$ necessarily have ...

Let $L$ be a finite non-abelian simple group, and let $D$ be a conjugacy class of involutions in $L$. Consider the complete graph $\Gamma$ with vertex...

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 $\ell(X)$ denote the composition length of a finite group $X$. Let $A$ be a finite nilpotent group acting by automorphisms on a finite soluble gro...

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...

A finite group $G$ is said to be semi-abelian if it has a sequence of subgroups $1=G_0\leqslant G_1\leqslant\cdots\leqslant G_n=G$ such that for every...

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?...

Let $\Gamma$ be a finite simple group and let $N_n(\Gamma)$ denote the set of normal subgroups of the free group $F_n$ of rank $n$ whose quotient is i...

Conjecture: For $n\geqslant 3$, there are no finite simple characteristic quotients of the free group $F_n$....

Conjecture: Metabelian groups are permutation-stable....

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?...

A group $G$ is said to be sofic if for every finite set $F\subseteq G$ containing $1$ and every $\varepsilon>0$ there exist $n\in\mathbb N$ and a map ...

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...

Conjecture: The sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$ is at least the $p'$-part of $|G|$....

Conjecture: The number of irreducible $p$-Brauer characters of a finite group $G$ is bounded above by the maximum of the number of conjugacy classes $...

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...

Conjecture: If $G$ is a transitive permutation group on a finite set $\Omega$, then for any distinct $\alpha,\beta\in\Omega$ there is an element $g\in...

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 $...