Nearrings and a construction of triply factorized groups [Elektronische Ressource] / vorgelegt von Peter Hubert

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• Mathematics

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2005
• Nombre de lectures: 9
• Langue: English

Extrait

Nearrings and a Construction
of Triply Factorized Groups
Dissertation
zur Erlangung des Grades
Doktor
”
der Naturwissenschaften“
am Fachbereich Physik, Mathematik und Informatik
der Johannes Gutenberg-Universit¨at
in Mainz
vorgelegt von
Peter Hubert
geboren am 27. Oktober 1972 in Wiesbaden
Mainz, im Juli 2005Tag der mundlichen Prufung: 25. Oktober 2005¨ ¨Summary
In this thesis a connection between nearrings and triply factorized groups is investigated. A
group G is called triply factorized by its subgroups A, B, and M, if G =AM =BM =AB,
where M is a normal subgroup of G and A∩M =B∩M = 1. Many problems in the theory
of factorized groups can be reduced to triply factorized groups (c.f. [2]).
Triply factorized groups are connected with radical rings in a natural way. A ringR is called
◦radical, ifR forms a groupR under the “circle operation”a◦b =ab+a+b for everya, b∈R.
◦ +In a radical ringR operates on the additive groupR and it can be shown that the semidirect
◦ + ◦productR nR is a grouptriplyfactorizedby twosubgroupsA andB isomorphic toR anda
+normal subgroupM isomorphic toR . Hence, in the triply factorized groups obtained in this
way, the normal subgroup M is always abelian. If, on the other hand, G =AM =BM =AB
is a triply factorized group with abelian subgroups A, B, and M and A∩B = 1, there exists
always a radical ring, from which G can be obtained as above (Sysak [20], see also [2]).
To construct triply factorized groups G = AM = BM = AB with non-abelian normal
subgroup M, a method using nearrings is described. Nearrings are a generalisation of rings
in the sense that the additive group of a nearring is not necessarily abelian and only one
distributive law holds. If R is a nearring with identity element 1 and U is a subgroup of the
+additive group R such that U +1 is a subgroup of the group of units of R, then U is called
a construction subgroup of R. For instance the Jacobson radical J(R) of any ring R is a subgroup of R. It is shown that U +1 operates on a construction subgroup U,
such that the semidirect product (U +1)nU is a group triply factorized by two subgroups A
and B isomorphic to U +1 and a normal subgroup isomorphic to U. Conversely, it is proved
that every triply factorized group G =AM =BM =AB with A∩B = 1 can be obtained by
a suitable nearring with this method. This generalises the above mentioned theorem of Sysak.
To know more about construction subgroups, the structure of nearrings is investigated in
detail. Here local nearrings, i.e. nearrings in which the set of all elements which are not right
invertible forms an additive group, play a special rˆole. In these nearrings the group of non-
invertible elements forms a construction subgroup. Given an arbitrary p-group N of finite
exponent (p a prime), a technique to construct local nearrings with a construction subgroup
that contains a subgroup isomorphic to N is developed.
Moreover, all triply factorized groups that can be constructed using a local nearring R of
3order p (p a prime) are described depending on the structure of the additive group of R.
4These triply factorized groups have orderp . There exist two different triply factorized groups
4of orderp for every primep. But it turns out that forp≥ 5 there is only one triply factorized
3 + 2group that can be constructed by a local nearring R of order p , if the exponent of R is p
+and R is not abelian.
Finally, all local nearrings R with dihedral group of units are classified. It turns out that
these nearrings are finite and their order does not exceed 16. It is shown that if R is such a
local nearring, then its additive group is ap-group forp = 2 orp = 3. This is done by showing
that no group of order 32 can occur as the additive group of a local nearring with dihedral
group of units. Some of the calculations in this classification were made with the computer
algebra system GAP. The programs used here are described in detail in Appendix B.Contents
Introduction 6
Notation 10
1. Triply factorized groups 15
1.1. Factorized groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2. A connection between triply factorized groups and radical rings . . . . . 16
2. Nearrings 18
2.1. Basics about nearrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2. Homomorphisms and modules . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1. Nearring homomorphisms . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2. Nearring modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3. Ideals and special subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1. Nearring and module ideals . . . . . . . . . . . . . . . . . . . . . 24
2.3.2. R-subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.3. Generalised annihilators . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.4. Properties of ideals and R-subgroups . . . . . . . . . . . . . . . . 28
3. Constructing triply factorized groups 29
3.1. Construction of triply factorized groups . . . . . . . . . . . . . . . . . . . 29
3.2. Triply factorized groups constructible by nearrings . . . . . . . . . . . . . 31
4. More on nearrings 36
4.1. Radical theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.1. Monogenic R-modules and modules of type ν . . . . . . . . . . . 36
4.1.2. Quasiregularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2. Nearfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3. Prime rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4. More on construction subgroups . . . . . . . . . . . . . . . . . . . . . . . 44
4.5. Subdirect sums and products . . . . . . . . . . . . . . . . . . . . . . . . 47
4Contents
5. Local nearrings 49
5.1. Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1.1. Definition of local nearrings . . . . . . . . . . . . . . . . . . . . . 49
5.1.2. Basic properties of local nearrings . . . . . . . . . . . . . . . . . . 50
+5.1.3. Properties of the additive group R . . . . . . . . . . . . . . . . . 52
5.1.4. The structure of L . . . . . . . . . . . . . . . . . . . . . . . . . . 54R
5.1.5. Simple local nearrings . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2. The structure of local nearrings . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.1. Prime rings of local nearrings . . . . . . . . . . . . . . . . . . . . 62
×5.2.2. The multiplicative group R . . . . . . . . . . . . . . . . . . . . . 64
5.2.3. The nearfield R/L . . . . . . . . . . . . . . . . . . . . . . . . . . 66R
5.3. Local nearrings with nilpotent L . . . . . . . . . . . . . . . . . . . . . . 69R
5.4. Subdirect products of local nearrings . . . . . . . . . . . . . . . . . . . . 70
6. An example for a local nearring 75
6.1. Construction of a local nearring R . . . . . . . . . . . . . . . . . . . . . . 75
6.2. The examination of the structure of R . . . . . . . . . . . . . . . . . . . 78
+6.2.1. The additive group R . . . . . . . . . . . . . . . . . . . . . . . . 78
×6.2.2. The multiplicative group R . . . . . . . . . . . . . . . . . . . . . 81
6.2.3. The operation of L +1 on L . . . . . . . . . . . . . . . . . . . 82R R
37. Triply factorized groups constructed by local nearrings of order p 84
7.1. Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
+7.2. The case R cyclic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
+7.3. The case R elementary abelian . . . . . . . . . . . . . . . . . . . . . . . 86
+ 27.4. The case R abelian of exponent p . . . . . . . . . . . . . . . . . . . . . 89
+7.5. The case R non-abelian of exponent p . . . . . . . . . . . . . . . . . . . 92
+ 27.6. The case Relian of exponent p . . . . . . . . . . . . . . . . . . 98
8. Local nearrings with dihedral multiplicative group 106
8.1. General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.2. Nearrings of odd order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.3.ings of even order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A. C++-Program used in Example 5.2.12 124
B. GAP-programs used in the proof of Theorem 8.3.11 139
Bibliography 145
5Introduction
A group G is called triply factorized if G =AM =BM =AB for two subgroups A and
B and a normal subgroup M of G where A∩M = B∩M = 1, i.e. G is a semidirect
product An M = B n M of A with M and of B with M. Many problems in the
theory of factorized groups can be reduced to questions about triply factorized groups
(c.f. Amberg, Franciosi, de Giovanni [2]). Therefore it is desirable to obtain examples of
such groups.
◦AringRiscalledradical,ifRisagroupR underthe“circleoperation”a◦b =ab+a+b
for all a, b∈R, or equivalently, if R coincides with its Jacobson radicalJ(R). If R is a
◦ +radical ring, the group R operates on the additive group R , such that the semidirect
◦ +product R nR is triply factorized. This observation was first described by Sysak [20]
(c.f. [2, Section 6.1]). If G = An M = B n M = AB is a triply factorized group
constructed with this method, the normal subgroup M is isomorphic to the additive
group of R and hence always is abelian (c.f. Construction 1.2.2).
Conversely,SysakprovedthatifG =AnM =BnM =AB isatriplyfactorizedgroup
with abelian subgroups A, B,