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 finite group G admits an automorphism \(\varphi \) of prime order coprime to |G| such that for some positive integer m every element of the centralizer \(C_G(\varphi )\) has a left Engel sink of cardinality at most m, then the index of the second Fitting subgroup \(F_2(G)\) is bounded in terms of m. 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 \([\ldots [[g,x],x],\dots ,x]\) 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 finite group G admits an automorphism \(\varphi \) of prime order coprime to |G| such that for some positive integer m every element of the centralizer \(C_G(\varphi )\) has a right Engel sink of cardinality at most m, then the index of the Fitting subgroup \(F_1(G)\) is bounded in terms of m.

组G的元素g的左恩格斯宿为集合\（{\ 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，正是当我们可以选择\（{\ mathscr {E}}（g）= \ {1 \} \）时，g才是左Engel元素。）我们证明，如果有限群G承认自同构\（\素数阶素数为|的varphi \）G | 这样对于某个正整数m扶正器的每个元素\（C_G（\ varphi）\）最多具有m个左恩格尔基数，则第二个Fitting子组\（F_2（G）\）的索引以m为界。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 \），所有足够长的换向器\（[\ ldots [[g，x]，x]，\ dots，x] \属于\（{\ mathscr {R}}（g）\）。（因此，正是当我们可以选择\（{\ 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承认自同构\（\ varphi \）的素数阶素数为| G | 这样，对于某个正整数m，扶正器\（C_G（\ varphi）\）的每个元素最多具有一个m的右恩格尔基数，然后将Fitting子组\（F_1（G）\）的索引限制在计米。