Given a finite group $G$, we denote by $ν(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $ν(G)>1/12,$ then $G$ is solvable.
We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that with the only specific exception the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.
We say a group $G$ has property $R_\infty$ if the number $R(φ)$ of twisted conjugacy classes is infinite for every automorphism $φ$ of $G$. For such groups, the $R_\infty$-nilpotency degree is the least integer $c$ such that $G/γ_{c+1}(G)$ has property $R_\infty$. In this work, we compute the $R_\infty$-nilpotency degree of all Generalized Solvable Baumslag-Solitar groups $Γ_n$. Moreover, we compute the lower central series of $Γ_n$, write the nilpotent quotients $Γ_{n,c}=Γ_n/γ_{c+1}(Γ_n)$ as semidirect products of finitely generated abelian groups and classify which integer invertible matrice…
We prove that every finitely generated, virtually solvable minimax group can be expressed as a homomorphic image of a virtually torsion-free, virtually solvable minimax group. This result enables us to generalize a theorem of Ch. Pittet and L. Saloff-Coste about random walks on finitely generated, virtually solvable minimax groups. Moreover, the paper identifies properties, such as the derived length and the nilpotency class of the Fitting subgroup, that are preserved in the covering process. Finally, we determine exactly which infinitely generated, virtually solvable minimax groups also posse…
We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. We show, in particular, that for every polynomial $r(t) = a_0 + a_1 \cdot t + \cdots + a_d \cdot t^d \in \mathbb{Z}[t]$ that is irreducible over $\mathbb{Q}$, there exist (explicit) invariants $a,b,c \in \mathbb{N}$ with the following property. Consider a finite group with a fix-point-free automorphism $α:{G}\longrightarrow{G}$ and suppose that for all $x \in G$ we have the equality $x^{a_0} \cdot α(x^{a_1})…
In the paper we characterize the class of finite solvable groups by two-variable identities in a way similar to the characterization of finite nilpotent groups by Engel identities. More precisely, a sequence of words $u_1,...,u_n,... $ is called correct if $u_k\equiv 1$ in a group $G$ implies $u_m\equiv 1$ in a group $G$ for all $m>k$. We are looking for an explicit correct sequence of words $u_1(x,y),...,u_n(x,y),...$ such that a group $G$ is solvable if and only if for some $n$ the word $u_n$ is an identity in $G$. Let $u_1=x^{-2}y\min x$, and $u_{n+1} = [xu_nx\min,yu_ny\min]$. The main resu…
We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class $n$ and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. We prove that central extensions of most finite simple groups have trivial Bogomolov multiplier.
We study relations between the additive and the multiplicative groups of a two-sided skew brace. In particular, we prove that if the additive group of a two-sided skew brace is finite solvable (respectively, finitely generated nilpotent, finitely generated residually nilpotent, finitely generated residually finite), then the multiplicative group of this skew brace is solvable (respectively, solvable, residually solvable, residually finite). Also, we prove that if the multiplicative group of a two-sided skew brace is nilpotent of nilpotency class $k$, then the additive group of this skew brace …
Given a finite bijective non-degenerate set-theoretic solution $(X,r)$ of the Yang--Baxter equation we characterize when its structure monoid $M(X,r)$ is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions $(X,r)$ and show, for example, that this property is equivalent to the solution associated to the structure monoid $M…
We study finite groups $G$ with the property that for any subgroup $M$ maximal in $G$ whose order is divisible by all the prime divisors of $|G|$, $M$ is supersolvable. We show that any nonabelian simple group can occur as a composition factor of such a group and that, if $G$ is solvable, then the nilpotency length and the rank are arbitrarily large. On the other hand, for every prime $p$, the $p$-length of such a group is at most $1$. This answers questions proposed by V. Monakhov in The Kourovka Notebook.
We study the groups $G$ with the curious property that there exists an element $k\in G$ and a function $f\colon G\to G$ such that $f(xk)=xf(x)$ holds for all $x\in G$. This property arose from the study of near-rings and input-output automata on groups. We call a group with this property a $J$-group. Finite $J$-groups must have odd order, and hence are solvable. We prove that every finite nilpotent group of odd order is a $J$-group if its nilpotency class $c$ satisfies $c\le6$. If $G$ is a finite $p$-group, with $p>2$ and $p^2>2c-1$, then we prove that $G$ is $J$-group. Finally, if $p>2$ and $…
We consider expansive group actions on a compact metric space containing a special fixed point denoted by $0$, and endomorphisms of such systems whose forward trajectories are attracted toward $0$. Such endomorphisms are called asymptotically nilpotent, and we study the conditions in which they are nilpotent, that is, map the entire space to $0$ in a finite number of iterations. We show that for a large class of discrete groups, this property of nil-rigidity holds for all expansive actions that satisfy a natural specification-like property and have dense homoclinic points. Our main result in p…
