Profinite groups and centralizers of coprime automorphisms whose elements are Engel

2017/07/20 by Acciarri, Cristina, da Silveira, Danilo Sanção
#20D45 #20E18 #20F40 #20F45 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1707.06889

Abstract

Let q be a prime, n a positive integer and A an elementary abelian group of order q^r with r≥2 acting on a finite q'-group G. The following results are proved. We show that if all elements in γ_r-1(C_G(a)) are n-Engel in G for any a∈ A^#, then γ_r-1(G) is k-Engel for some \n,q,r\-bounded number k, and if, for some integer d such that 2^d≤ r-1, all elements in the dth derived group of C_G(a) are n-Engel in G for any a∈ A^#, then the dth derived group G^(d) is k-Engel for some \n,q,r\-bounded number k. Assuming r≥ 3 we prove that if all elements in γ_r-2(C_G(a)) are n-Engel in C_G(a) for any a∈ A^#, then γ_r-2(G) is k-Engel for some \n,q,r\-bounded number k, and if, for some integer d such that 2^d≤ r-2, all elements in the dth derived group of C_G(a) are n-Engel in C_G(a) for any a∈ A^#, then the dth derived group G^(d) is k-Engel for some \n,q,r\-bounded number k. Analogue (non-quantitative) results for profinite groups are also obtained.

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