Related papers: On groups with large verbal quotients

We show that, there exists a constant $a$ such that, for every subgroup $H$ of a finite group $G$, the number of maximal subgroups of $G$ containing $H$ is bounded above by $a|G:H|^{3/2}$. In particular, a transitive permutation group of…

We prove that outer commutator words are uniformly concise, i.e. if an outer commutator word w takes m different values in a group G, then the order of the verbal subgroup w(G) is bounded by a function depending only on m and not on w or G.…

If w is a word in d>1 letters and G is a finite group, evaluation of w on a uniformly randomly chosen d-tuple in G gives a random variable with values in G, which may or may not be uniform. It is known that if G ranges over finite simple…

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every…

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup…

We study maximal subsemigroups of the monoid T(X) of all full transformations on the set X=N of natural numbers containing a given subsemigroup W of T(X) where each element of a given set $U$ is a generator of T(X) modulo W. This note…

We will give an example of a branch group $G$ that has exponential growth but does not contain any non-abelian free subgroups. This answers question 16 from \cite{Bartholdi} positively. The proof demonstrates how to construct a non-trivial…

Let $d$ be a positive integer. A finite group is called $d$-maximal if it can be generated by precisely $d$ elements, while its proper subgroups have smaller generating sets. For $d\in\{1,2\}$, the $d$-maximal groups have been classified up…

Let m, n be positive integers, v a multilinear commutator word and w = v^m. Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of all groups G in which the w-values are…

Let $G$ be a $p$-group for some prime $p$. Let $n$ be the positive integer so that $|G:Z(G)| = p^n$. Suppose $A$ is a maximal abelian subgroup of $G$. Let $$p^l = {\rm max} \{|Z(C_G (g)):Z(G)| : g \in G \setminus Z(G)\},$$ $$p^b = {\rm max}…

Let $\gamma_k=[x_1,\dots,x_k]$ be the $k$-th lower central group-word. Given a group $G$, we write $X_k(G)$ for the set of $\gamma_k$-values and $\gamma_k(G)$ for the $k$-th term of the lower central of $G$. This paper deals with groups in…

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…

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_k = \langle x_1, x_2, ..., x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i < j \rangle$. So $G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z}…

Let $H$ be an abelian subgroup of a finite group $G$ and $\pi$ the set of prime divisors of $|H|$. We prove that $|H O_{\pi}(G)/ O_{\pi}(G)|$ is bounded above by the largest character degree of $G$. A similar result is obtained when $H$ is…

A word $w$ in a free group is {\em achiral} if for every group $G,$ $G_w=G_{w^{-1}},$ where $G_w$ is the image of the word map $w$ on $G.$ We will give few classes of examples of achiral words. Cocke and Ho asked whether Engel words are…

Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. By a result of Glauberman it has been known for a long time that if $G$ is nilpotent of class 2, then $|G| < |V|$. In this paper we generalize this result…

We show, in particular, that, if a finite group $H$ is a retract of any finite group containing $H$ as a verbally closed subgroup, then the centre of $H$ is a direct factor of $H$.

Sabatini (2024) defined a subgroup $H$ of $G$ to be an exponential subgroup if $x^{|G:H|} \in H$ for all $x \in G$. Exponential subgroups are a generalization of normal (and subnormal) subgroups: all subnormal subgroups are exponential, but…

If $g\in G$ is a non-trivial element in a residually finite group, then there exists by definition a finite group $Q$ and a homomorphism $\varphi: G \to Q$ such that $\varphi(g) \neq e$. The residual finiteness growth $\text{RF}_G$ of a…

In this note we prove that if $G$ is a finitely generated profinite group then the verbal subgroup $G^{q}$ is open. Equivalently in a $d$-generator finite group every product of $q$th powers is a product of $f(d,q)$ $q$th powers.