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POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)<sub>โ</sub> GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN
Abstract Let G be a polycyclic, metabelian or soluble of type (FP) โ 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.