Which Words have an Efficient Result?
LI Qian-lu · Communications in Algebra · 2004
Black [Black, S. (1999 Black, S. 1999. Which words spell “almost nilpotent?”. J. Algebra, 221: 475–496. [Crossref], [Web of Science ®] , [Google Scholar]). Which words spell “almost nilpotent?” J. Algebra 221: 475–496] defined a class of words which are efficient or which have an efficient result. The variety of residually finite groups defined by a word of this class, like that defined by a positive word, is nilpotent-of-bounded-class-by-bounded-exponent. We introduce a notion of a standard exponent for a homogeneous word and we show that a homogeneous word has an efficient result if and only if its standard exponent is 1. With this we shall improve the main results of Black (1999 Black, S. 1999. Which words spell “almost nilpotent?”. J. Algebra, 221: 475–496. [Crossref], [Web of Science ®] , [Google Scholar]) including its theorem. We also answer its first two open questions.