61 research outputs found

### Commutators and commutator subgroups in profinite groups

Let $G$ be a profinite group. We prove that the commutator subgroup $G'$ is finite-by-procyclic if and only if the set of all commutators of $G$ is contained in a union of countably many procyclic subgroups.Comment: 19 pages, final versio

### Double automorphisms of graded Lie algebras

We introduce the concept of a double automorphism of an A-graded Lie algebra L. Roughly, this is an automorphism of L which also induces an automorphism of the group A. It is clear that the set of all double automorphisms of L forms a subgroup in Aut(L). In the present paper we prove several nilpotency criteria for a graded Lie algebra admitting a finite group of double automorphisms. We also give an application of our results to groups admitting a Frobenius group of automorphisms.Comment: 13 page

### On finite groups in which coprime commutators are covered by few cyclic subgroups

The coprime commutators $\gamma_j^*$ and $\delta_j^*$ were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders. They are defined as follows. Let $G$ be a finite group. Every element of $G$ is both a $\gamma_1^*$-commutator and a $\delta_0^*$-commutator. Now let $j\geq 2$ and let $X$ be the set of all elements of $G$ that are powers of $\gamma_{j-1}^*$-commutators. An element $g$ is a $\gamma_j^*$-commutator if there exist $a\in X$ and $b\in G$ such that $g=[a,b]$ and $(|a|,|b|)=1$. For $j\geq 1$ let $Y$ be the set of all elements of $G$ that are powers of $\delta_{j-1}^*$-commutators. The element $g$ is a $\delta_j^*$-commutator if there exist $a,b\in Y$ such that $g=[a,b]$ and $(|a|,|b|)=1$. The subgroups of $G$ generated by all $\gamma_j^*$-commutators and all $\delta_j^*$-commutators are denoted by $\gamma_j^*(G)$ and $\delta_j^*(G)$, respectively. For every $j\geq2$ the subgroup $\gamma_j^*(G)$ is precisely the last term of the lower central series of $G$ (which throughout the paper is denoted by $\gamma_\infty(G)$) while for every $j\geq1$ the subgroup $\delta_j^*(G)$ is precisely the last term of the lower central series of $\delta_{j-1}^*(G)$, that is, $\delta_j^*(G)=\gamma_\infty(\delta_{j-1}^*(G))$. In the present paper we prove that if $G$ possesses $m$ cyclic subgroups whose union contains all $\gamma_j^*$-commutators of $G$, then $\gamma_j^*(G)$ contains a subgroup $\Delta$, of $m$-bounded order, which is normal in $G$ and has the property that $\gamma_{j}^{*}(G)/\Delta$ is cyclic. If $j\geq2$ and $G$ possesses $m$ cyclic subgroups whose union contains all $\delta_j^*$-commutators of $G$, then the order of $\delta_j^*(G)$ is $m$-bounded.Comment: Final version, referee's suggestions adde
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