On Certain Weak Engel-Type Conditions in Groups

Abstract

Let w(x, y) be a word in two variables and 𝔚 the variety determined by w. In this paper we raise the following question: if for every pair of elements a, b in a group G there exists g ∈ G such that w(a ^g , b) = 1, under what conditions does the group G belong to 𝔚? In particular, we consider the n-Engel word w(x, y) = [x, _n y]. We show that in this case the property is satisfied when the group G is metabelian. If n = 2, then we extend this result to the class of all solvable groups.

Key Words:

• Conjugate
• Engel conditions
• Solvable groups
• Two-variable word

2010 Mathematics Subject Classification:

• 20E25
• 20F12
• 20F16
• 20F45

Notes

Communicated by A. Olshanskii.