For an element g of a group G, an Engel sink is a subset E (g) such that for every x ∈ G all sufficiently long commutators [x,g,g,...,g] belong to E(g). Let q be a prime, let m be a positive integer and A an elementary abelian group of order q^2 acting coprimely on a finite group G. We show that if for each nontrivial element a in A and every element g ∈ C_G(a) the cardinality of the smallest Engel sink E (g) is at most m, then the order of γ∞(G) is bounded in terms of m only. Moreover we prove that if for each a ∈ A {1} and every element g ∈ CG(a), the smallest Engel sink E(g) generates a subgroup of rank at most m, then the rank of γ∞(G) is bounded in terms of m and q only.

Engel sinks of fixed points in finite groups / Acciarri, C; Shumyatsky, P; Silveira, D. S.. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - 223:11(2019), pp. 4592-4601. [https://doi.org/10.1016/j.jpaa.2019.02.006]

Engel sinks of fixed points in finite groups

Acciarri C;
2019-01-01

Abstract

For an element g of a group G, an Engel sink is a subset E (g) such that for every x ∈ G all sufficiently long commutators [x,g,g,...,g] belong to E(g). Let q be a prime, let m be a positive integer and A an elementary abelian group of order q^2 acting coprimely on a finite group G. We show that if for each nontrivial element a in A and every element g ∈ C_G(a) the cardinality of the smallest Engel sink E (g) is at most m, then the order of γ∞(G) is bounded in terms of m only. Moreover we prove that if for each a ∈ A {1} and every element g ∈ CG(a), the smallest Engel sink E(g) generates a subgroup of rank at most m, then the rank of γ∞(G) is bounded in terms of m and q only.
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Engel sinks of fixed points in finite groups / Acciarri, C; Shumyatsky, P; Silveira, D. S.. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - 223:11(2019), pp. 4592-4601. [https://doi.org/10.1016/j.jpaa.2019.02.006]
Acciarri, C; Shumyatsky, P; Silveira, D. S.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1255515
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