On a problem of P. Hall for Engel words |
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Authors: | Alireza Abdollahi Francesco G Russo |
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Institution: | 1.Department of Mathematics,University of Isfahan,Isfahan,Iran;2.School of Mathematics,Institute for Research in Fundamental Sciences (IPM),Tehran,Iran;3.Dipartimento di Matematica e Informatica,Universitá degli Studi di Palermo,Palermo,Italy;4.Department of Mathematics,Universiti Teknologi Malaysia,Skudai,Malaysia |
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Abstract: | Let θ be a word in n variables and let G be a group; the marginal and verbal subgroups of G determined by θ are denoted by θ( G) and θ *( G), respectively. The following problems are generally attributed to P. Hall: - (I)
If π is a set of primes and |G : θ *(G)| is a finite π-group, is θ(G) also a finite π-group? - (II)
If θ(G) is finite and G satisfies maximal condition on its subgroups, is |G : θ *(G)| finite? - (III)
If the set \({\{\theta(g_1,\ldots,g_n) | g_1,\ldots,g_n\in G\}}\) is finite, does it follow that θ(G) is finite?
We investigate the case in which θ is the n-Engel word e n = x, n y] for \({n\in\{2,3,4\}}\) . |
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