Artin Transfer Patterns on Descendant Trees of Finite p-Groups

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.