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

G组元素g的右Engel汇是一个\（{{\ mathscr {R}}}（g）\）集合，使得对于每个\（x \ in G \），所有足够长的换向子\（[ ... [[[g，x]，x]，\ dots，x] \）属于\（{\ mathscr {R}}（g）\）。（因此，恰恰当我们可以选择\（{{\ mathscr {R}}}（g）= \ {1 \} \时，g才是正确的恩格尔元素。）我们证明，如果有限群G允许共素自同构。\（\ varphi \）的素数阶的，使得每一个固定点\（\ varphi \）具有有限的右安水槽，然后ģ有一个开放的本地幂等子组。G组元素g的左Engel汇是集合\（{{\ mathscr {E}}}（g）\），使得对于每个\（x \ in G \），所有足够长的换向器\（[ ... [[[x，g]，g]，\ dots，g] \属于\（{{\ mathscr {E}}}（g）\）。（因此，恰好在我们可以选择\（{\ mathscr {E}}（g）= \ {1 \} \时，g才是左Engel元素。）我们证明，如果有限群G承认共质子自同构\（素数阶\ varphi \），使得\（\ varphi \）的每个固定点都有一个有限的左恩格尔接收器，然后是G 有一个开放的逐幂子组。