International Conferences 2016
July 25  27, 2016:
2nd International Conference
on Groups and Algebras
ICGA 2016, Suzhou
Presentation:
pCapitulation over Number Fields
with pClass Rank Two,
Suzhou,
China
November 11  12, 2016:
International Colloquium of Algebra,
Number Theory, Cryptography,
and Information Security
ANCI 2016, Taza
Invited Lecture:
Recent Progress in Determining
pClass Field Towers,
Taza,
Morocco

Definitions
Let G be a group with a subgroup H < G of finite index (G:H) = n.
Suppose that a left transversal of H in G is given by
G = g(1)H + … + g(n)H.
Then the left action of a group element x in G causes a permutation of the left cosets
xg(i)H = g(s(i))H, for i in {1,…,n},
with a permutation s in the symmetric group S(n) of degree n.
The Artin transfer T(G,H): G → H/H' from G to the abelianization H/H' of H
is a group homomorphism defined by
T(G,H)(x) = g(s(1))^{1}xg(1)* … * g(s(n))^{1}xg(n) * H'.
In particular, if p is a prime number and G is a finite pgroup or an infinite prop group
then we denote by Lyr(1,G) the first layer of normal subgroups H < G with index (G:H) = p in G.
If G has a finite abelianization G/G' then Lyr(1,G) is a finite set
and we call
the family of transfer targets (abelianizations)
t(G) = ( H/H' ) with H in Lyr(1,G)
the Indexp Abelianization Data, briefly the IPAD of G, and
the family of transfer kernels
k(G) = ( Ker(T(G,H) ) with H in Lyr(1,G)
the Indexp Obstruction Data, briefly the IPOD of G.
The pair A(1,G) = ( t(G),k(G) ) is the first layer of the Artin pattern A(G) of G.
Examples
Towers of Type H.4
Towers of Type G.19
Background and Aims
The Artin pattern A(G) of a prop group G has turned out to be the decisive information for solving
the problem of the Hilbert pclass field tower F(p,∞,K) of an algebraic number field K.
Within the frame of our
International Scientific Research Project
with title
Towers of pclass fields over algebraic number fields,
which is supported financially by the Austrian Science Fund (FWF):
P 26008N25
,
this goal of identifying the pclass tower F(p,∞,K) has been approached in several steps.

The metabelianization G/G'' of the Galois group G of the pclass tower,
which describes the lowest two stages of the tower,
K < F(p,1,K) < F(p,2,K),
was investigated in four subsequent articles, forming a Tetralogy,
[1] The second pclass group of a number field, 2012
,
[2] Transfers of metabelian pgroups, 2012
,
[3] Principalization algorithm via class group structure, 2014
,
[4] The distribution of second pclass groups on coclass graphs, 2013
.
In [3] and [4], the IPAD was called the transfer target type, briefly TTT,
and the IPOD was called the transfer kernel type, briefly TKT.

Inspired by our cooperation with Michael R. Bush and the resulting joint paper
3class field towers of exact length 3, 2015
,
we pushed forward beyond the twostage towers into the strange realm of
nonmetabelian pgroups with derived length 3,
K < F(p,1,K) < F(p,2,K) < F(p,3,K),
whose investigation requires
iterated IPADs of second order developed systematically in a Trilogy,
[5] Periodic bifurcations in descendant trees of finite pgroups, 2015
,
[6] Indexp abelianization data of pclass tower groups, 2015
,
[7] Artin transfer patterns on descendant trees of finite pgroups, 2016
,
with parallel applications to special types of algebraic number fields in the following articles,
[8] Periodic sequences of pclass tower groups, 2015
,
[9] New number fields with known pclass tower, 2016
,
and in cooperation with Abdelmalek Azizi, Abdelkader Zekhnini and Mohammed Taous,
[10] Principalization of 2class groups of type (2,2,2) of biquadratic fields, 2015
,
respectively with Abdelmalek Azizi, Mohamed Talbi, Mohammed Talbi, and Aissa Derhem,
[11] The group Gal( K_{3}^{2} / K ) for K = Q( (3)^{1/2}, D^{1/2} ) of type (3,3), 2016
.


International Research Project
Principal investigator of the
International Research Project
Towers of pClass Fields
over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
P26008N25
Time Schedule:
