On Certain Weak Engel-Type Conditions in Groups |
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Authors: | Maurizio Meriano Chiara Nicotera |
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Institution: | 1. Dipartimento di Matematica , Università di Salerno , Fisciano , Salerno , Italy mmeriano@unisa.it;3. Dipartimento di Matematica , Università di Salerno , Fisciano , Salerno , Italy |
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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. |
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Keywords: | Conjugate Engel conditions Solvable groups Two-variable word |
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