IPAD and IPOD

Index-p Abelianization Data

Index-p Obstruction Data

International Conferences 2016 July 25 - 27, 2016: 2nd International Conference on Groups and Algebras ICGA 2016, Suzhou Presentation: p-Capitulation over Number Fields with p-Class 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 p-Class Field Towers, Taza, Morocco Our services to the mathematical community: Mathematical Reviews (MR) 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))-1xg(1)* … * g(s(n))-1xg(n) * H'. In particular, if p is a prime number and G is a finite p-group or an infinite pro-p 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 Index-p 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 Index-p 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 pro-p group G has turned out to be the decisive information for solving the problem of the Hilbert p-class field tower F(p,∞,K) of an algebraic number field K. Within the frame of our International Scientific Research Project with title Towers of p-class fields over algebraic number fields, which is supported financially by the Austrian Science Fund (FWF): P 26008-N25 , this goal of identifying the p-class tower F(p,∞,K) has been approached in several steps. The metabelianization G/G'' of the Galois group G of the p-class 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 p-class group of a number field, 2012 , [2] Transfers of metabelian p-groups, 2012 , [3] Principalization algorithm via class group structure, 2014 , [4] The distribution of second p-class 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 3-class field towers of exact length 3, 2015 , we pushed forward beyond the two-stage towers into the strange realm of non-metabelian p-groups 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 p-groups, 2015 , [6] Index-p abelianization data of p-class tower groups, 2015 , [7] Artin transfer patterns on descendant trees of finite p-groups, 2016 , with parallel applications to special types of algebraic number fields in the following articles, [8] Periodic sequences of p-class tower groups, 2015 , [9] New number fields with known p-class tower, 2016 , and in cooperation with Abdelmalek Azizi, Abdelkader Zekhnini and Mohammed Taous, [10] Principalization of 2-class 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( K32 / K ) for K = Q( (-3)1/2, D1/2 ) of type (3,3), 2016 . International Research Project Principal investigator of the International Research Project Towers of p-Class Fields over Algebraic Number Fields supported by the Austrian Science Fund (FWF): P26008-N25 Time Schedule: FWF Project P26008-N25