Words and Almost Nilpotent Varieties of Groups<sup>#</sup>

Li, Qianlu

doi:10.1080/00927870500243239

Cited by 2 publications

(3 citation statements)

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“…The referee has pointed out that Qianlu Li [9,Theorem 23] has shown that if a word "spells virtual nilpotence," then it has an "efficient result", i.e., a consequence in a certain special sense, satisfying Sarah Black's criterion (2). The above proof seems rather simpler than Li's, although on the face of it not quite so general.…”

Section: Corollary 1 the Sufficient Condition Of Sarah Black's Theormentioning

confidence: 99%

“…), which side of the above dichotomy it falls on. Such an algorithm has been given by Qianlu Li in [8,9], which, although not formulated explicitly in terms of the wreath product W := C C, again begs to be so formulated, in our opinion.…”

Section: Corollary 1 the Sufficient Condition Of Sarah Black's Theormentioning

confidence: 99%

“…We infer in particular that her condition is also necessary. We also deduce a simplified version of an algorithm of Qianlu Li [8,9] for deciding whether or not a given law w(x 1 , . .…”

Section: Introductionmentioning

confidence: 99%

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

Burns

Macedońska

Medvedev

2008

Communications in Algebra

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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. In [2] Sarah Black 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 [4] for arbitrary words w(x 1 , . . . , x n ), encompasses Black's condition, extending it to the nvariable case and a certain large class S (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 Qianlu Li [8,9] for deciding whether or not a given law w(x 1 , . . . , x n ) ≡ 1 satisfies the extended version of Black's criterion.

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Section: Corollary 1 the Sufficient Condition Of Sarah Black's Theormentioning

confidence: 99%

Section: Corollary 1 the Sufficient Condition Of Sarah Black's Theormentioning

confidence: 99%