Let q be a prime and A an elementary abelian q-group acting as a coprime group of automorphisms on a profinite group G. We show that if A is of order q^2 and some power of each element in C_G(a) is Engel in G for any a ∈ A^#, then G is locally virtually nilpotent. Assuming that A is of order q^3, we prove that if some power of each element in C_G(a) is Engel in C_G(a) for any a ∈ A^#, then G is locally virtually nilpotent. Some analogues of quantitative nature for finite groups are also obtained.
Engel-like conditions in fixed points of automorphisms of profinite groups / Acciarri, C; Silveira, D. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 199:1(2020), pp. 187-197. [10.1007/s10231-019-00872-7]
Engel-like conditions in fixed points of automorphisms of profinite groups
Acciarri C
;
2020
Abstract
Let q be a prime and A an elementary abelian q-group acting as a coprime group of automorphisms on a profinite group G. We show that if A is of order q^2 and some power of each element in C_G(a) is Engel in G for any a ∈ A^#, then G is locally virtually nilpotent. Assuming that A is of order q^3, we prove that if some power of each element in C_G(a) is Engel in C_G(a) for any a ∈ A^#, then G is locally virtually nilpotent. Some analogues of quantitative nature for finite groups are also obtained.
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