Abstract
Based on a thorough theory of the Artin transfer homomorphism from a group G to the abelianization of a subgroup of finite index , and its connection with the permutation representation and the monomial representation of G, the Artin pattern , which consists of families , resp. , of transfer targets, resp. transfer kernels, is defined for the vertices of any descendant tree T of finite p-groups. It is endowed with partial order relations and , which are compatible with the parent-descendant relation of the edges of the tree T. The partial order enables termination criteria for the p-group generation algorithm which can be used for searching and identifying a finite p-group G, whose Artin pattern is known completely or at least partially, by constructing the descendant tree with the abelianization of G as its root. An appendix summarizes details concerning induced homomorphisms between quotient groups, which play a crucial role in establishing the natural partial order on Artin patterns and explaining the stabilization, resp. polarization, of their components in descendant trees T of finite p-groups.
Highlights
In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index
Such transfer mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin’s reciprocity isomorphism ([1], §4, Allgemeines Reziprozitätsgesetz, p. 361) to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups ([2], §2, p. 50)
The reason is that the unramified extensions of a base field contain information in the shape of capitulation patterns and class group structures, and these arithmetic invariants can be translated into group theoretic data on transfer kernels and targets by means of Artin’s reciprocity law of class field theory
Summary
Independently of number theoretic applications, a natural partial order on the kernels and targets of Artin transfers, has recently been found to be compatible with parent-child relations between finite p-groups, where p denotes a prime number. Artin transfers provide valuable information for classifying finite p-groups by kernel-target patterns and for searching and identifying particular groups in descendant trees by looking for patterns defined by kernels and targets of Artin transfers These strategies of pattern recognition are useful in purely group theoretic context, and, most importantly, for applications in algebraic number theory concerning Galois groups of higher p-class fields and Hilbert p-class field towers. Inspired by Bourbaki’s method of exposition [14], Appendix on induced homomorphisms, which is separated to avoid a disruption of the flow of exposition, goes down to the origins exploiting set theoretic facts concerning direct images and inverse pre-images of mappings which are crucial for explaining the natural partial order of Artin patterns
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