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On the existence of Engel pairs in certain linear groups

S. G. Quek

Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

E-mail Address: quekshg@yahoo.com

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K. B. Wong

Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

E-mail Address: kbwong@um.edu.my

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P. C. Wong

Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

E-mail Address: wongpc@um.edu.my

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https://doi.org/10.1142/S0219498816500626Cited by:0 (Source: Crossref)

Abstract

Let $G$ be a group and $h, g \in G$ . The 2-tuple $(h, g)$ is said to be an $n$ -Engel pair, $n \geq 2$ , if $h = [h,_{n} g]$ , $g = [g,_{n} h]$ and $h \neq 1$ . Let SL $(2, F)$ be the special linear group of degree $2$ over the field $F$ . In this paper, we show that given any field $L$ , there is a field extension $F$ of $L$ with $[F : L] \leq 6$ such that SL $(2, F)$ has an $n$ -Engel pair for some integer $n \geq 4$ . We will also show that SL $(2, F)$ has a $5$ -Engel pair if $F$ is a field of characteristic $p \equiv \pm 1 mod 5$ .

Communicated by I. M. Isaacs

Keywords:

AMSC: 20F05, 20F45, 20B30

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