Minimal presentations of free metabelian nilpotent groups

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Iraghi Moghaddam, Gholamhossein
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Minimal presentations of free metabelian nilpotent groups, in terms of basic commutators, are investigated. For 'm, n' >= 2, let ' M'('m,n') be a free metabelian nilpotent group of rank ' m' and of nilpotency class 'n' - 1. In Chapter 2 we have shown that for 'n' = 2,3,4, 'M'(' m,n') admits a minimal presentation whose set of defining relators is the set of all basic commutators of weight 'n'; this is in fact a yes answer for these values of 'n' to the question raised by Charles C. Sims in this regard. In Chapter 3 the same result is obtained for 'M'(2,5). For 'm' = 2 and 'n' >= 6 in Chapter 3 we have found a minimal presentation of 'M'(2,'n') with the set of relators consisting of certain types of basic commutators of weight at most 'n'. Finally for 'm' >= 3 and 'n' >= 5, first in Section 2 of Chapter 2 we present a finite presentation of ' M'('m,n'), and then in Chapter 4 we refine this presentation to a sharper one. In Chapter 5 we offer a last refinement and introduce a very sharp presentation of 'M'('m,n'). All of the results are obtained using only pure group theoretical techniques without involving any computer methods.