Finite groups with a soluble group of coprime automorphisms whose fixed points have bounded Engel sinks. (English. Russian original) Zbl 07789142
Algebra Logic 62, No. 1, 80-93 (2023); translation from Algebra Logika 62, No. 1, 114-134 (2023).
Summary: Suppose that a finite group \(G\) admits a soluble group of coprime automorphisms A. We prove that if, for some positive integer \(m\), every element of the centralizer \(C_G(A)\) has a left Engel sink of cardinality at most \(m\) (or a right Engel sink of cardinality at most \(m\)), then \(G\) has a subgroup of \((|A|,m)\)-bounded index which has Fitting height at most \(2 \alpha (A) + 2\), where \(\alpha (A)\) is the composition length of \(A\). We also prove that if, for some positive integer \(r\), every element of the centralizer \(C_G(A)\) has a left Engel sink of rank at most \(r\) (or a right Engel sink of rank at most \(r)\), then \(G\) has a subgroup of \((|A|, r)\)-bounded index which has Fitting height at most \(4 \alpha(A) + 4 \alpha(A) + 3\). Here, a left Engel sink of an element \(g\) of a group \(G\) is a set \(\mathfrak{E} (g)\) such that for every \(x \in G\) all sufficiently long commutators \([\dots[[x, g], g], \dots, g]\) belong to \(\mathfrak{E} (g)\). (Thus, \(g\) is a left Engel element precisely when we can choose \(\mathcal{E}(g) = \{1\}\).) A right Engel sink of an element \(g\) of a group \(G\) is a set \(\mathcal{R}(g)\) such that for every \(x \in G\) all sufficiently long commutators \([\dots[[g, x], x], \dots, x]\) belong to \(\mathcal{R}(g)\). Thus, \(g\) is a right Engel element precisely when we can choose \(\mathcal{R}(g) = \{1\}\).
MSC:
| 20D45 | Automorphisms of abstract finite groups |
| 20F45 | Engel conditions |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
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