Coprime automorphisms of finite groups

Let $G$ be a finite group admitting a coprime automorphism $alpha$ of order $e$. Denote by $I_G(alpha)$ the set of commutators $g^{-1}g^alpha$, where $gin G$, and by $[G,alpha]$ the subgroup generated by $I_G(alpha)$. We study the impact of $I_G(alpha)$ on the structure of $[G,alpha]$. Suppose that each subgroup generated by a subset of $I_G(alpha)$ can be generated by at most $r$ elements. We show that the rank of $[G,alpha]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(alpha)$ has odd order, then $[G,alpha]$ has odd order too. Further, if every pair of elements from $I_G(alpha)$ generates a soluble, or nilpotent, subgroup, then $[G,alpha]$ is soluble, or respectively nilpotent.

Profinite groups with restricted centralizers of $pi$-elements

A group $G$ is said to have restricted centralizers if for each $g$ in $G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a set of primes $pi$, we take interest in profinite groups with restricted centralizers of $pi$-elements. It is shown that such a profinite group has an open subgroup of the form $Ptimes Q$, where $P$ is an abelian pro-$pi$ subgroup and $Q$ is a pro-$pi$ subgroup. This significantly strengthens a result from our earlier paper.

A stronger form of Neumanns BFC-theorem

Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing the element $x$. A famous theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the derived group $G$ is finite. We establish the following result. Let $n$ be a positive integer and $K$ a subgroup of a group $G$ such that $|x^G|leq n$ for each $xin K$. Let $H=langle K^Grangle$ be the normal closure of $K$. Then the order of the derived group $H$ is finite and $n$-bounded. Some corollaries of this result are also discussed.

On groups in which Engel sinks are cyclic

For an element $g$ of a group $G$, an Engel sink is a subset $mathcal{E}(g)$ such that for every $ xin G $ all sufficiently long commutators $ [x,g,g,ldots,g] $ belong to $mathcal{E}(g)$. We conjecture that if $G$ is a profinite group in which every element admits a sink that is a procyclic subgroup, then $G$ is procyclic-by-(locally nilpotent). We prove the conjecture in two cases -- when $G$ is a finite group, or a soluble pro-$p$ group.

The generating graph of the abelian groups

For a group $G,$ let $Gamma(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Moreover let $Gamma^*(G)$ be the subgraph of $Gamma(G)$ that is induced by all the vertices of $Gamma(G)$ that are not isolated. We prove that if $G$ is a 2-generated non-cyclic abelian group then $Gamma^*(G)$ is connected. Moreover $mathrm{diam}(Gamma^*(G))=2$ if the torsion subgroup of $G$ is non-trivial and $mathrm{diam}(Gamma^*(G))=infty$ otherwise. If $F$ is the free group of rank 2, then $Gamma^*(F)$ is connected and we deduce from $mathrm{diam}(Gamma^*(mathbb{Z}times mathbb{Z}))=infty$ that $mathrm{diam}(Gamma^*(F))=infty.$

Engel sinks of fixed points in finite groups

For an element $g$ of a group $G$, an Engel sink is a subset $mathscr{E}(g)$ such that for every $ xin G $ all sufficiently long commutators $ [x,g,g,ldots,g] $ belong to $mathscr{E}(g)$. Let $q$ be a prime, let $m$ be a positive integer and $A$ an elementary abelian group of order $q^2$ acting coprimely on a finite group $G$. We show that if for each nontrivial element $a$ in $ A$ and every element $gin C_{G}(a)$ the cardinality of the smallest Engel sink $mathscr{E}(g)$ is at most $m$, then the order of $gamma_infty(G)$ is bounded in terms of $m$ only. Moreover we prove that if for each $ain Asetminus {1}$ and every element $gin C_{G}(a)$, the smallest Engel sink $mathscr{E}(g)$ generates a subgroup of rank at most $m$, then the rank of $gamma_infty(G)$ is bounded in terms of $m$ and $q$ only.

Graphs encoding the generating properties of a finite group

Assume that $G$ is a finite group. For every $a, b inmathbb N,$ we define a graph $Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^acup G^b$ and in which two tuples $(x_1,dots,x_a)$ and $(y_1,dots,y_b)$ are adjacent if and only if $langle x_1,dots,x_a,y_1,dots,y_b rangle =G.$ We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about $G$ are encoded by these graphs.

Profinite groups and centralizers of coprime automorphisms whose elements are Engel

Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $rgeq2$ acting on a finite $q$-group $G$. The following results are proved. We show that if all elements in $gamma_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $ain A^#$, then $gamma_{r-1}(G)$ is $k$-Engel for some ${n,q,r}$-bounded number $k$, and if, for some integer $d$ such that $2^dleq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $ain A^#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some ${n,q,r}$-bounded number $k$. Assuming $rgeq 3$ we prove that if all elements in $gamma_{r-2}(C_G(a))$ are $n$-Engel in $C_G(a)$ for any $ain A^#$, then $gamma_{r-2}(G)$ is $k$-Engel for some ${n,q,r}$-bounded number $k$, and if, for some integer $d$ such that $2^dleq r-2$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $C_G(a)$ for any $ain A^#,$ then the $d$th derived group $G^{(d)}$ is $k$-Engel for some ${n,q,r}$-bounded number $k$. Analogue (non-quantitative) results for profinite groups are also obtained.

Profinite groups and the fixed points of coprime automorphisms

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 $ain A^{#}$. Then the group $G$ is locally nilpotent.

On verbal subgroups in finite and profinite groups

Let $w$ be a multilinear commutator word. In the present paper we describe recent results that show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely (or in some cases countably) many subgroups with a prescribed property, then the verbal subgroup $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank.