1. Foundations of 2Stage Metabelian 3Groups.DEFINITION 1.1.For any two rational integers n >= m >= 2, we denote by ZEF(m,n) the set of all 2stage metabelian 3groups G of class m1 and order 3^{n} with commutator factor group G/G' = (3,3). Remarks. 1. Nebelung's designation ZEF ("Rang Zwei oder Eins Faktoren") refers to the property that all factors G_{i}/G_{i+1} of the descending central series G = G_{1} > G_{2} > ... > G_{m1} > G_{m} = 1, which is defined by G_{i+1} = [G_{i},G] for all i >= 1, are of elementary abelian type, either (3,3) or (3). 2. The groups of maximal class with m = n, where only cyclic factors (3) occur, coincide with Blackburn's groups CF(m,n,p) of maximal class with m = n and p = 3: ZEF(m,m) = CF(m,m,3). 3. A less local, i. e., more international, designation for ZEF would be CBF ... Cyclic or Bicyclic Factor groups. 

2. Structure of Higher Members of the Descending Central Series.Preliminaries.For a nonnegative integer n, denote by A(3,n) the abelian 3group of type (3^{q+r},3^{q}), where n = 2q + r with unique q >= 0 and 0 <= r <= 1 ("almost homogeneous" abelian 3groups). THEOREM 2.1. Assumptions: Let m,n be rational integers, such that 3 <= m <= n <= 2m  3, and put e = n  m + 2. Let G be a 2stage metabelian 3group in ZEF(m,n). Claims: 1. If G is of maximal class, i. e., m = n resp. e = 2, then the descending central series exclusively consists of members having factors G_{i}/G_{i+1} = (3) of 3rank 1 (symbol  ), G_{i} = A(3,mi) for all 3 <= i <= m.
2. If G is not of maximal class, i. e., 4 <= m < n resp. e >= 3, then the descending central series begins with members having factors G_{i}/G_{i+1} = (3,3) of 3rank 2 (symbol  ), G_{i} = A(3,mi) * A(3,e+1i) for all 3 <= i <= e and ends with members having factors G_{i}/G_{i+1} = (3) of 3rank 1 (symbol  ), G_{i} = A(3,mi) for all e+1 <= i <= m.


3. Structure of the (abelian) Commutator Subgroup.The statements about the higher members of the descending central seriesin the previous theorem start with G_{3}, since the results on the Commutator Subgroup G' = G_{2} include an irregular case. THEOREM 3.1. Assumptions: Let m,n be rational integers, such that 3 <= m <= n <= 2m  3, and put e = n  m + 2. Let G be a 2stage metabelian 3group in ZEF(m,n). Claims: 1. If G is of maximal class, i. e., m = n resp. e = 2, then G_{2} = A(3,m2). 2. If G is not of maximal class, i. e., 4 <= m < n resp. e >= 3, and if G = G^{(m,n)}((a,b,c,d),r) with relational exponents 1 <= a,b,c,d,r <= 1, then a) in the irregular case G in ZEF 1b(m,n) with odd m and n = 2m  4, i. e., e = m  2, and 0 != r = b  1 for m = 5, and r = 1 for m > 5, we have G_{2} = A(3,m3) * A(3,m3) (where m  3 = e  1) b) and otherwise (regular case) always G_{2} = A(3,m2) * A(3,e2). 

4. Characteristic Subgroups between G and G'.The theorems in the preceding 2 sections revealedthe structure of all abelian members of the descending central series of a 2stage metabelian 3group. In certain cases, however, there is also an interesting intermediate group between G_{1} and G_{2}, i. e., above the commutator subgroup: DEFINITION 4.1. 1. For any positive rational integer i >= 1, we denote by C_{i} the subgroup of G with the property that C_{i}/G_{i+2} coincides with the centralizer of G_{i}/G_{i+2} in G/G_{i+2}, i. e., C_{i} consists of all elements g in G, such that all commutators [g,u] with u in G_{i} are contained in G_{i+2}. 2. Let s = min{ 1 <= i <= m  C_{i} > G' }. Remarks. 1. The C_{i} ( i >= 1) are ascending characteristic subgroups of G that contain G'. 2. For m = 2, we have n = 2 and G' = 1 < C_{i} = G = (3,3) for all i >= 1, i. e., s = 1. 3. For 3 <= m <= n, the C_{i} are partitioned in the following manner: G' = C_{1} = ... = C_{s1} < C_{s} = ... = C_{m2} < C_{m1} = G, i. e., 2 <= s <= m  1. DEFINITION 4.2. For any rational integers n >= m >= 3, we define a disjoint partition ZEF(m,n) = ZEF 1(m,n) + ZEF 2(m,n), where for the groups G in ZEF 1(m,n) = { G in ZEF(m,n)  s = m1 } none of the characteristic subgroups C_{i} lies between G and G', whereas the groups G in ZEF 2(m,n) = { G in ZEF(m,n)  s < m1 } show the behavior in the diagram below. 

5. Generators and Relations.This will be the topic of our next article. 


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