1. Commutators Generating the Descending Central Series.PROPOSITION 1.1.Assumptions: Let m,n be rational integers, such that 3 <= m <= n and let G be a 2stage metabelian 3group in ZEF(m,n) with generators x,y. Claims: 1. G_{2} = < [y,x],G_{3} > , i. e., G_{2}/G_{3} = (3) is cyclic 2. G_{2} = < [y,x]^{g}  g in G > = [y,x]^{G} , i. e., G_{2} is a cyclic G/G'module 3. For all i >= 2, we have the following implication: if G_{i} = < g_{1},...,g_{r},G_{i+1} > with r >= 0 and g_{1},...,g_{r} in G_{i}  G_{i+1}, then r <= 2^{i1} and G_{i+1} = < [g_{j},x],[g_{j},y],G_{i+2}  1 <= j <= r > 4. In particular, if 4 <= m <= n and if we use the notation s_{2} = [y,x], s_{3} = [s_{2},x] = s_{2}^{x1}, t_{3} = [s_{2},y] = s_{2}^{y1}, then G_{2} = < s_{2},G_{3} > implies that G_{3} = < s_{3},t_{3},G_{4} > 5. For 4 <= m <= n, there exist generators x,y of G, such that G_{3} = < x^{3},y^{3},G_{4} > . 

2. Groups of Maximal Class.THEOREM 2.1.Assumptions: Let m be a rational integer, such that m >= 3, and let G be a 2stage metabelian 3group of maximal class in ZEF(m,m) with generators s,s_{1}, where we assume s in G  C_{2} and s_{1} in C_{2}  G' in the case of m > 3, i. e., G' < C_{2} < G. Finally, with d(i) = (s  1)^{i} let s_{i} = [s_{1},s]^{d(i2)} in G_{i} for i >= 2. Claims: 1. The generators satisfy the following relations: a) relations without s,s_{1}: s_{i} = 1 for i >= m, [s_{i},s_{j}] = 1 for i,j >= 2, s_{i}^{3}s_{i+1}^{3}s_{i+2} = 1 for i >= 2 b) relations including s_{1} but without s: [s_{i},s_{1}] = 1 for i >= 3, [s_{2},s_{1}] = s_{m1}^{c} for some 1 <= c <= 1, s_{1}^{3}s_{2}^{3}s_{3} = s_{m1}^{b} for some 1 <= b <= 1 c) relations including s: [s_{i},s] = s_{i+1} for i >= 1, s^{3} = s_{m1}^{a} for some 1 <= a <= 1. These relations are abbreviated by writing G = G^{(m)}(a,b,c). 2. The structure of the members of the descending central series is given by G_{i} = < s_{i} > * < s_{i+1} > = A(3,mi) for all 2 <= i <= m1, in particular G_{m1} = < s_{m1} > = (3).


3. Groups of NonMaximal Class.THEOREM 3.1.Assumptions: Let m,n be rational integers, such that 4 <= m < n <= 2m  3, and put e = n  m + 2 >= 3. Let G be a 2stage metabelian 3group of nonmaximal class in ZEF(m,n) with generators x,y, such that G_{3} = < x^{3},y^{3},G_{4} >, where we assume x in G  C_{s} and y in C_{s}  G' in the case of s < m  1, i. e., G in ZEF 2(m,n). Additionally, with v_{2} = [y,x], d(x,i) = (x  1)^{i}, and d(y,i) = (y  1)^{i} let v_{i} = [y,x]^{d(x,i2)}, w_{i} = [y,x]^{d(y,i2)}, for i >= 3, s_{i} = (y^{3})^{d(x,i3)}, t_{i} = (x^{3})^{d(y,i3)}, for i >= 3, S_{i} = < s_{j}  j >= i >, T_{i} = < t_{j}  j >= i >, for i >= 3, and denote by D the intersection of S_{4} with T_{4}. Claims: 1. G_{i} = < s_{i}, t_{i}, G_{i+1} > = S_{i}*T_{i} for i >= 3, and thus G_{i}/G_{i+1} is at most of 3rank 2. 2. S_{i} = < s_{i} > * < s_{i+1} > ~ A(3,mi) and #S_{i} = 3^{mi} for all 3 <= i <= m. 3. If G belongs to ZEF a(m,n), then we have a) T_{i} = < t_{i} > * < t_{i+1} > ~ A(3,e+1i) and #T_{i} = 3^{e+1i} for all 3 <= i <= e+1, b) the intersection of S_{i} with T_{i} is trivial (=1), c) and G_{i} is the direct product of S_{i} and T_{i}: G_{i} = < s_{i} > * < s_{i+1} > * < t_{i} > * < t_{i+1} > ~A(3,mi)*A(3,e+1i) for 3 <= i <= e and ~A(3,mi) for e+1 <= i <= m. 4. However, if G belongs to ZEF b(m,n), then a) T_{i} = < t_{i} > * < t_{i+1} > ~ A(3,e+2i) and #T_{i} = 3^{e+2i} for all 3 <= i <= e+2, b) and the intersection of S_{i} with T_{i} equals G_{m1} = < s_{m1} > = < t_{e+1} > for all 3 <= i <= e+1. 

4. Transfers ("Verlagerungen").This will be the topic of our next article. 


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