Related papers: Rank type conditions on commutators in finite grou…

A finite group $G$ is coprimely-invariably generated if there exists a set of generators $\{g_1, \ldots, g_d\}$ of $G$ with the property that the orders $|g_1|, \ldots, |g_d|$ are pairwise coprime and that for all $x_1, \ldots, x_d \in G$…

Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group…

We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w-values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G) is locally finite. If G is…

Suppose that $G$ is a finite, transitive, solvable permutation group acting on a set $S$ with $n$ elements. Let $G_0$ be the stabilizer of a point $\alpha \in \Omega$. Define the rank of a permutation group, denoted $r(G),$ as the number of…

Assume $G$ is a finite $p$-group, and let $S$ be a Sylow $p$-subgroup of $\operatorname{Aut}(G)$ with $\exp(S)=q$. We prove that if $G$ is of class $c$, then $\exp(G)|p^{\ceil{\log_pc}}q^3$, and if $G$ is a metabelian $p$-group of class at…

Given two subgroups H,K of a finite group G, the probability that a pair of random elements from H and K commutes is denoted by \pr(H,K). We address the following question. Let P be a p-subgroup of a finite group G and assume that…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $\lambda (G)$ as the minimum number of nonsoluble factors in a series of…

We study finitely generated models of countable theories, having at most countably many nonisomorphic finitely generated models. We intro- duce a notion of rank of finitely generated models and we prove, when T has at most countably many…

Let $P,$ $S,$ and $T$ be semigroups, $f:P\to S$ and $g:P\to T$ semigroup homomorphisms, and $X$ a generating set for $S$ (possibly infinite). Clearly, a <i>necessary</i> condition for there to exist a homomorphism $S\to T$ making a…

A group $G$ is invariably generated (IG) if there is a subset $S \subseteq G$ such that for every subset $S' \subseteq G$, obtained from $S$ by replacing each element with a conjugate, $S'$ generates $G$. $G$ is finitely invariably…

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite…

The Pr\"ufer rank $\mathrm{rk}(G)$ of a profinite group $G$ is the supremum, across all open subgroups $H$ of $G$, of the minimal number of generators $\mathrm{d}(H)$. It is known that, for any given prime $p$, a profinite group $G$ admits…

Let N be a normal subgroup of a finite group G. We prove that under certain (unavoidable) conditions the subgroup [N,G] is a product of commutators [N,y] (with prescribed values of y from a given set Y) of length bounded by a function of…

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq \epsilon >…

Let $G$ be a group. An automorphism of $G$ is called intense if it sends each subgroup of $G$ to a conjugate; the collection of such automorphisms is denoted by $\mathrm{Int}(G)$. In the special case in which $p$ is a prime number and $G$…

Let $G$ be a finite group, let $x \in G$, and let $p$ be a prime. We prove that the commutator $[x,g]$ is a $p$-element for every $g \in G$ if and only if $x$ is central modulo $\mathbf{O}_p(G)$, where $\mathbf{O}_p(G)$ denotes the largest…

Given a finitely generated subgroup G of a ring R we provide a finite subset of G such that if each element of this set satisfies some cubic polynomial equation in one variable over the center Z of R then the subring generated by G has…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, assume that $G$ has a maximal $A$-invariant subgroup $M$ that is a direct product of some isomorphic simple groups, we prove that if $G$ has a…

Let $G$ be a group and $H \le K \le G$. We say that $H$ is $c$-embedded in $G$ with respect to $K$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H \cap B \le Z(K)$. Given a finite group $G$, a prime number $p$ and a Sylow…