 # 2-Stage Metabelian 3-Groups

## 1. Foundations of 2-Stage Metabelian 3-Groups.

DEFINITION 1.1.
For any two rational integers n >= m >= 2,
we denote by ZEF(m,n)
the set of all 2-stage metabelian 3-groups G
of class m-1 and order 3n
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 Gi/Gi+1
of the descending central series G = G1 > G2 > ... > Gm-1 > Gm = 1,
which is defined by Gi+1 = [Gi,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 non-negative integer n, denote by A(3,n)
the abelian 3-group of type (3q+r,3q),
where n = 2q + r with unique q >= 0 and 0 <= r <= 1
("almost homogeneous" abelian 3-groups).

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 2-stage metabelian 3-group 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 Gi/Gi+1 = (3) of 3-rank 1 (symbol | ),
Gi = A(3,m-i) for all 3 <= i <= m.

 Gm = 1 | Gm-1 | ... | G4 | G3 | G2 / / \ \ M1 M2 M3 M4 \ \ / / G1 = G

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 Gi/Gi+1 = (3,3) of 3-rank 2 (symbol || ),
Gi = A(3,m-i) * A(3,e+1-i) for all 3 <= i <= e
and ends with
members having factors Gi/Gi+1 = (3) of 3-rank 1 (symbol | ),
Gi = A(3,m-i) for all e+1 <= i <= m.

 Gm = 1 | Gm-1 | ... | Ge+2 | Ge+1 || Ge || ... || G4 || G3 | G2 / / \ \ M1 M2 M3 M4 \ \ / / G1 = G

## 3. Structure of the (abelian) Commutator Subgroup.

The statements about the higher members of the descending central series
since the results on the Commutator Subgroup G' = G2
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 2-stage metabelian 3-group in ZEF(m,n).

Claims:
1. If G is of maximal class,
i. e., m = n resp. e = 2, then
G2 = A(3,m-2).

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
G2 = A(3,m-3) * A(3,m-3) (where m - 3 = e - 1)

b) and otherwise (regular case) always
G2 = A(3,m-2) * A(3,e-2).

## 4. Characteristic Subgroups between G and G'.

The theorems in the preceding 2 sections revealed
the structure of all abelian members of the descending central series
of a 2-stage metabelian 3-group.
In certain cases, however, there is also an interesting intermediate group
between G1 and G2, i. e., above the commutator subgroup:

DEFINITION 4.1.
1. For any positive rational integer i >= 1,
we denote by Ci the subgroup of G
with the property that Ci/Gi+2 coincides with the centralizer
of Gi/Gi+2 in G/Gi+2,
i. e., Ci consists of all elements g in G, such that
all commutators [g,u] with u in Gi are contained in Gi+2.
2. Let s = min{ 1 <= i <= m | Ci > G' }.

Remarks.
1. The Ci ( i >= 1) are ascending characteristic subgroups of G that contain G'.
2. For m = 2, we have n = 2 and G' = 1 < Ci = G = (3,3) for all i >= 1, i. e., s = 1.
3. For 3 <= m <= n, the Ci are partitioned in the following manner:
G' = C1 = ... = Cs-1 < Cs = ... = Cm-2 < Cm-1 = 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 = m-1 }
none of the characteristic subgroups Ci lies between G and G',

 G = Cm-1 | M | G' = C1 = ... = Cm-2

whereas the groups G in
ZEF 2(m,n) = { G in ZEF(m,n) | s < m-1 }
show the behavior in the diagram below.

 G = Cm-1 | M = Cs = ... = Cm-2 | G' = C1 = ... = Cs-1

## 5. Generators and Relations.

This will be the topic of our next article.
 References:  Arnold Scholz und Olga Taussky, Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper, J. reine angew. Math. 171 (1934), 19 - 41  Norman Blackburn, On a special class of p-groups, Acta Math. 100 (1958), 45 - 92  Brigitte Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Köln, 1989  Daniel C. Mayer, Principalization in complex S3-fields, Congressus Numerantium 80 (1991), 73 - 87

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