POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)(infinity) GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN

摘要

Let G be a polycyclic, metabelian or soluble of type (FP)(infinity) group such that the class Rat(G) of all rational subsets of G is a Boolean algebra. Then, G is virtually abelian. Every soluble biautomatic group is virtually abelian.