Burnside metabelian groups

Abstract

If B is a group of prime-power exponent p e and solubility class 2, then B has nilpotency class at most e ( p e — p e-1 )+1 provided the number of generators of B are at most p +1. Representa­tions of B are constructed which in the case of two generators and prime exponent is a faithful representation of the free group of the variety under study and for prime-power exponent show the existence of a group with nilpotency class e ( p e — p e-1 ). In the general situation where B as above has exponent n , and n is not a prime-power, the place where the lower central series of G becomes stationary is determined by a knowledge of the nilpotency class of the groups of prime-power exponent for all prime divisors of n . The bound e ( p e — p e-1 )+1 on the nilpotency class is a consequence of the following: Let G be a direct product of at most p —1 cyclic groups of order p e and R the group ring of G over the integers modulo p e . Then the e ( p e — p e-1 ) th power of the augmentation ideal of R is contained in the ideal of R generated by all 'cyclotomic’ polynomials Ʃ p e -1 i = 0 g i for all g in G . If G is a direct product of more than p +1 cyclic groups, then this result is no longer true unless e = 1.