Centennial 2004



Fundamental facts concerning

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
//\\
M1M2M3M4
\\//
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
//\\
M1M2M3M4
\\//
G1 = G

3. Structure of the (abelian) Commutator Subgroup.

The statements about the higher members of the descending central series
in the previous theorem start with G3,
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:

[1] Arnold Scholz und Olga Taussky,
Die Hauptideale der kubischen Klassenkörper
imaginär quadratischer Zahlkörper,
J. reine angew. Math. 171 (1934), 19 - 41

[2] Norman Blackburn,
On a special class of p-groups,
Acta Math. 100 (1958), 45 - 92

[3] Brigitte Nebelung,
Klassifikation metabelscher 3-Gruppen
mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem,
Inauguraldissertation, Köln, 1989

[4] Daniel C. Mayer,
Principalization in complex S3-fields,
Congressus Numerantium 80 (1991), 73 - 87

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