arXiv subjects
Publications and source records attributed to Cristina Acciarri.
The main result of the paper is the following theorem. Let $q$ be a prime and $A$ an elementary abelian group of order $q^3$. Suppose that $A$ acts coprimely on a profinite group $G$ and assume that $C_G(a)$ is locally nilpotent for each $a\in A^{\#}$. Then the group $G$ is locally nilpotent.
We complete the study of finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let $q$ be a prime and $A$ an elementary abelian $q$-group of order at least $q^2$ acting coprimely on a profinite group $G$. Assume that all elements in $C_{G}(a)$ are Engel in $G$ for each $a\in A^{\#}$. Then $G$ is locally nilpotent (Theorem B2). Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^3$ acting coprimely on a finite group $G$. Assume that for each $a\in A^{\#}$ every element of $C_{G}(a)$ is $n$-Engel in $C_{G}(a)$. Then the group $G$ is $k$-Engel for some $\{n,q\}$-bounded number $k$ (Theorem A3).
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.
Let $w$ be a group-word. Suppose that the set of all $w$-values in a profinite group $G$ is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup $w(G)$ depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to relation of the described results with Hall's problem on conciseness of group-words.
The coprime commutators $γ_j^*$ and $δ_j^*$ 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 $γ_1^*$-commutator and a $δ_0^*$-commutator. Now let $j\geq 2$ and let $X$ be the set of all elements of $G$ that are powers of $γ_{j-1}^*$-commutators. An element $g$ is a $γ_j^*$-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 $δ_{j-1}^*$-commutators. The element $g$ is a $δ_j^*$-commutator if there exist $a,b\in Y$ such that $g=[a,b]$ and $(|a|,|b|)=1$. The subgroups of $G$ generated by all $γ_j^*$-commutators and all $δ_j^*$-commutators are denoted by $γ_j^*(G)$ and $δ_j^*(G)$, respectively. For every $j\geq2$ the subgroup $γ_j^*(G)$ is precisely the last term of the lower central series of $G$ (which throughout the paper is denoted by $γ_\infty(G)$) while for every $j\geq1$ the subgroup $δ_j^*(G)$ is precisely the last term of the lower central series of $δ_{j-1}^*(G)$, that is, $δ_j^*(G)=γ_\infty(δ_{j-1}^*(G))$. In the present paper we prove that if $G$ possesses $m$ cyclic subgroups whose union contains all $γ_j^*$-commutators of $G$, then $γ_j^*(G)$ contains a subgroup $Δ$, of $m$-bounded order, which is normal in $G$ and has the property that $γ_{j}^{*}(G)/Δ$ is cyclic. If $j\geq2$ and $G$ possesses $m$ cyclic subgroups whose union contains all $δ_j^*$-commutators of $G$, then the order of $δ_j^*(G)$ is $m$-bounded.
Let $G$ be a finite group. We show that the order of the subgroup generated by coprime $γ_k$-commutators (respectively $δ_k$-commutators) is bounded in terms of the size of the set of coprime $γ_k$-commutators (respectively $δ_k$-commutators). This is in parallel with the classical theorem due to Turner-Smith that the words $γ_k$ and $δ_k$ are concise.
A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group $G\in X$, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime-power, then the word $w^q$ is indeed concise in the class of residually finite groups. Further, we show that in the case where $w=γ_{k}$ the word $w^q$ is boundedly concise in the class of residually finite groups. It remains unknown whether the word $w^q$ is actually concise in the class of all groups.
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.
Let $G$ be a finite group of order $p^am$, where $p$ is a prime and $m$ is not divisible by $p$, and let $P$ be a Sylow $p$-subgroup of $G$. If $w$ is an outer commutator word, we prove that $P\cap w(G)$ is generated by the intersection of $P$ with the set of $m$th powers of all values of $w$ in $G$
For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $γ_k$-values in $G$, it is shown that $γ_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contains all $γ_k$-values, it is shown that $γ_k(G)$ has finite $(k,r,s)$-bounded rank.
Let $A$ be an elementary abelian group of order $p^{k}$ with $k\geq 3$ acting on a finite $p'$-group $G$. The following results are proved. If $γ_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then $γ_{k-2}(G)$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. If, for some integer $d$ such that $2^{d}+2\leq k$, the $d$th derived group of $C_{G}(a)$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. Earlier this was known only in the case where $k\leq 4$.
If G is a finitely generated powerful pro-p group satisfying a certain law v=1, and if G can be generated by a normal subset T of finite width which satisfies a positive law, we prove that G is nilpotent. Furthermore, the nilpotency class of G can be bounded in terms of the prime p, the number of generators of G, the law v=1, the width of T, and the degree of the positive law. The main interest of this result is the application to verbal subgroups: if G is a p-adic analytic pro-p group in which all values of a word w satisfy positive law, and if the verbal subgroup w(G) is powerful, then w(G) is nilpotent.