Bounding the exponent of a verbal subgroup
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- 作者：Eloisa Detomi (1)
Marta Morigi (2)
Pavel Shumyatsky (3) - 关键词：Verbal subgroup ; Commutator words ; Engel words ; Exponent ; 20F10 ; 20F45 ; 20F14
- 刊名：Annali di Matematica Pura ed Applicata
- 出版年：2014
- 出版时间：October 2014
- 年：2014
- 卷：193
- 期：5
- 页码：1431-1441
- 全文大小：177 KB
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Marta Morigi (2)
Pavel Shumyatsky (3)
1. Dipartimento di Matematica, Universit脿 di Padova, Via Trieste 63, 35121, Padova, Italy
2. Dipartimento di Matematica, Universit脿 di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy
3. Department of Mathematics, University of Brasilia, 70910-900, Brasilia, DF, Brazil - ISSN：1618-1891
文摘
We deal with the following conjecture. If \(w\) is a group word and \(G\) is a finite group in which any nilpotent subgroup generated by \(w\) -values has exponent dividing \(e\) , then the exponent of the verbal subgroup \(w(G)\) is bounded in terms of \(e\) and \(w\) only. We show that this is true in the case where \(w\) is either the \(n\text{ th }\) Engel word or the word \([x^n,y_1,y_2,\ldots ,y_k]\) (Theorem A). Further, we show that for any positive integer \(e\) there exists a number \(k=k(e)\) such that if \(w\) is a word and \(G\) is a finite group in which any nilpotent subgroup generated by products of \(k\) values of the word \(w\) has exponent dividing \(e\) , then the exponent of the verbal subgroup \(w(G)\) is bounded in terms of \(e\) and \(w\) only (Theorem B).