Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

A right Engel sink of an element g of a group G is a set $${{\mathscr {R}}}(g)$$ R ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[g,x],x],\dots ,x]$$ [ . . . [ [ g , x ] , x ] , ⋯ , x ] belong to $${\mathscr {R}}(g)$$ R ( g ) . (Thus, g is a right Engel element precisely when we can choose $${{\mathscr {R}}}(g)=\{ 1\}$$ R ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set $${{\mathscr {E}}}(g)$$ E ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[x,g],g],\dots ,g]$$ [ . . . [ [ x , g ] , g ] , ⋯ , g ] belong to $${{\mathscr {E}}}(g)$$ E ( g ) . (Thus, g is a left Engel element precisely when we can choose $${\mathscr {E}}(g)=\{ 1\}$$ E ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.

On Finite Groups with an Automorphism of Prime Order Whose Fixed Points Have Bounded Engel Sinks

A left Engel sink of an elementgof a groupGis a set$${\mathscr {E}}(g)$$E(g)such that for every$$x\in G$$x∈Gall sufficiently long commutators$$[...[[x,g],g],\dots ,g]$$[...[[x,g],g],⋯,g]belong to$${\mathscr {E}}(g)$$E(g). (Thus,gis a left Engel element precisely when we can choose$${\mathscr {E}}(g)=\{ 1\}$$E(g)={1}.) We prove that if a finite groupGadmits an automorphism$$\varphi $$φof prime order coprime to |G| such that for some positive integermevery element of the centralizer$$C_G(\varphi )$$CG(φ)has a left Engel sink of cardinality at mostm, then the index of the second Fitting subgroup$$F_2(G)$$F2(G)is bounded in terms ofm. A right Engel sink of an elementgof a groupGis a set$${\mathscr {R}}(g)$$R(g)such that for every$$x\in G$$x∈Gall sufficiently long commutators$$[\ldots [[g,x],x],\dots ,x]$$[…[[g,x],x],⋯,x]belong to$${\mathscr {R}}(g)$$R(g). (Thus,gis a right Engel element precisely when we can choose$${\mathscr {R}}(g)=\{ 1\}$$R(g)={1}.) We prove that if a finite groupGadmits an automorphism$$\varphi $$φof prime order coprime to |G| such that for some positive integermevery element of the centralizer$$C_G(\varphi )$$CG(φ)has a right Engel sink of cardinality at mostm, then the index of the Fitting subgroup$$F_1(G)$$F1(G)is bounded in terms ofm.

Compact groups in which all elements have countable right Engel sinks

A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every x ∈ G all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$ . (Thus, g is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$ .) It is proved that if every element of a compact (Hausdorff) group G has a countable right Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

Locally finite p-groups with a left 3-Engel element whose normal closure is not nilpotent

For any odd prime [Formula: see text], we give an example of a locally finite [Formula: see text]-group [Formula: see text] containing a left 3-Engel element [Formula: see text] where [Formula: see text] is not nilpotent.

Compact groups with countable Engel sinks

An Engel sink of an element [Formula: see text] of a group [Formula: see text] is a set [Formula: see text] such that for every [Formula: see text] all sufficiently long commutators [Formula: see text] belong to [Formula: see text]. (Thus, [Formula: see text] is an Engel element precisely when we can choose [Formula: see text].) It is proved that if every element of a compact (Hausdorff) group [Formula: see text] has a countable (or finite) Engel sink, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. This settles a question suggested by J. S. Wilson.

RIGHT ENGEL-TYPE SUBGROUPS AND LENGTH PARAMETERS OF FINITE GROUPS

Let $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the subgroup generated by all the right Engel values $[g,_{n}x]$ over $x\in G$. In the case when $G$ is soluble we prove that if, for some $n$, the Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble $G$, it is proved that if, for some $n$, the generalized Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the generalized Fitting subgroup $F_{f(k,m)}^{\ast }(G)$ with $f(k,m)$ depending only on $k$ and $m$, where $|g|$ is the product of $m$ primes counting multiplicities. It is also proved that if, for some $n$, the nonsoluble length of $R_{n}(g)$ is equal to $k$, then $g$ belongs to a normal subgroup whose nonsoluble length is bounded in terms of $k$ and $m$. Earlier, similar generalizations of Baer’s theorem (which states that an Engel element of a finite group belongs to the Fitting subgroup) were obtained by the first two authors in terms of left Engel-type subgroups.

Compact groups in which all elements are almost right Engel

We say that an element g of a group G is almost right Engel if there is a finite set R(g) such that for every x∈G, there is a positive integer n(x,g) such that […[[g,x],x],…,x]∈R(g) if x is repeated at least n(x,g) times. Thus, g is a right Engel element precisely when we can choose R(g)={1}. We prove that if all elements of a compact (Hausdorff) group G are almost right Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound ∣R(g)∣⩽m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson–Zelmanov theorem saying that profinite Engel groups are locally nilpotent and previous results of the authors about compact groups in which all elements are almost left Engel.