Profinite groups and centralizers of coprime automorphisms whose elements are Engel
Cristina Acciarri, Danilo San\c{c}\~ao da Silveira

TL;DR
This paper investigates the structure of finite and profinite groups with automorphisms of coprime order, showing that certain Engel conditions on centralizers imply bounded Engel properties on derived and lower central series.
Contribution
It establishes new bounds on Engel properties of groups based on conditions on centralizers under coprime automorphisms, extending to profinite groups.
Findings
Bounded Engel properties for derived subgroups under centralizer conditions
Results apply to both finite and profinite groups
Provides explicit bounds depending on parameters q, r, n
Abstract
Let be a prime, a positive integer and an elementary abelian group of order with acting on a finite -group . The following results are proved. We show that if all elements in are -Engel in for any , then is -Engel for some -bounded number , and if, for some integer such that , all elements in the th derived group of are -Engel in for any , then the th derived group is -Engel for some -bounded number . Assuming we prove that if all elements in are -Engel in for any , then is -Engel for some -bounded number , and if, for some integer such that , all elements in the th derived group of…
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