Related papers: Groups with a covering condition on commutators

We show that, if $w_1, \ldots , w_6$ are words which are not an identity of any (non-abelian) finite simple group, then $w_1(G)w_2(G) \cdots w_6(G) = G$ for all (non-abelian) finite simple groups $G$. In particular, for every word $w$,…

For a non-cyclic finite group $G$, let $\gamma(G)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently…

A cover of a unital, associative (not necessarily commutative) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the cardinality…

Let ${\cal N}_c$ be the variety of nilpotent groups of class at most $c\ \ (c\geq 2)$ and $G=Z_r\oplus Z_s $ be the direct sum of two finite cyclic groups. It is shown that if the greatest common divisor of $r$ and $s$ is not one, then $G$…

It is shown that, for any function $g$ that is weakly increasing on compact convex sets and has the property that if $\lambda\ge 0$ and $K^\prime$ is a translate of $\lambda K$ then $g(K^\prime) =\lambda g(K)$, then for any covering…

Let $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ be two $n$-tuples of positive integers, let $X$ be a set of positive integers, and let $g$ be a positive integer. In this work we show an algorithmic process in order to compute all the sets…

We provide a general sufficient condition for extendability of quasimorphisms on subgroups. This condition recovers the result of Hull--Osin on quasimorphisms on hyperbolically embedded subgroups, and the proof given in this paper is much…

For G = SL(3,R) and G = SO(2,n), we give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan…

In "All p-adic reductive groups are tame" Bernstein proved that for a reductive group G over a local non-archimedean field F and a compact open subgroup K of G there exists a uniform bound N(G,K) such that for every irreducible, smooth, and…

We show that if a finite simple group G isomorphic to PSL(n,q) or PSU(n,q), where either $n\ne 4$, or q is prime or even, acts on a vector space over a field of the defining characteristic of G, then the corresponding semidirect product…

For a p-group G admitting an automorphism $\phi$ of order $p^n$ with exactly $p^m$ fixed points such that $\phi^{p^{n-1}}$ has exactly $p^k$ fixed points, we prove that G has a fully-invariant subgroup of m-bounded nilpotency class with…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…

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…

In this thesis we study the following property of a finite group G: the minimal number n such that G embeds in Sn. We start with an explicit formula for the number n for abelian groups. Then, we study the behavior of this group property in…

We prove that the symmetric group $S_n$ has a unique minimal cover $\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the order of $\mathcal{M}$. In addition, we prove that the order of…

An $(n,k)$ sequence covering array is a set of permutations of $[n]$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of at least one of the permutations. An $(n,k)$ sequence covering array is perfect if there is a…

A group law is said to be detectable in power subgroups if, for all coprime $m$ and $n$, a group $G$ satisfies the law if and only if the power subgroups $G^m$ and $G^n$ both satisfy the law. We prove that for all positive integers $c$,…

Two groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$. In this paper, we give new necessary conditions for two groups to…

We say that a cover of surfaces S -> X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the…

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…