How to Tell if a Group Law Entails Virtual Nilpotence: Postscript

摘要

It is known that for any finitely generated group G from the large class of “locally graded” groups, satisfaction of an Engel or positive law forces G to be virtually nilpotent. Black (19991. Black , S. ( 1999 ). Which words spell “almost nilpotent”? J. Algebra 221 : 475 - 496 .[CrossRef], [Web of Science ®]View all references) gives a sufficient condition for an arbitrary 2-variable law w(x, y)≡ 1 to imply virtual nilpotence—though only for finitely generated residually finite groups. We show how the Dichotomy Theorem from Burns and Medvedev (20032. Burns , R. G. , Medvedev , Y. ( 2003 ). Group laws implying virtual nilpotence . J. Aust. Math. Soc. 74 : 295 - 312 .[CrossRef], [Web of Science ®]View all references) for arbitrary words w(x ₁,…,x _n ), encompasses Black's condition, extending it to the n-variable case and a certain large class (however still falling short of the class of locally graded groups). We infer in particular that her condition is also necessary. We also deduce a simplified version of an algorithm of Li (20047. Li , Q. ( 2004 ). Which words have an efficient result? Commun. Algebra 32 : 2527 - 2545 .[Taylor & Francis Online], [Web of Science ®]View all references, 20058. Li , Q. ( 2005 ). Words and almost nilpotent varieties of groups . Commun. Algebra 33 : 3569 - 3582 .[Taylor & Francis Online], [Web of Science ®]View all references) for deciding whether or not a given law w(x ₁,…,x _n )≡ 1 satisfies the extended version of Black's criterion.View full textDownload full textKey WordsGroup law, Variety, Virtual nilpotence, Word, Wreath product2000 Mathematics Subject Classification20E10, 20E22, 20F10Related var addthis_config = { ui_cobrand: "Taylor & Francis Online", services_compact: "citeulike,netvibes,twitter,technorati,delicious,linkedin,facebook,stumbleupon,digg,google,more", pubid: "ra-4dff56cd6bb1830b" }; Add to shortlist Link Permalink http://dx.doi.org/10.1080/00927870701776771