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On Semidirect Products and the Arithmetic Lifting Property
Author(s) -
Black Elena V.
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461079900784x
Subject(s) - mathematics , semidirect product , group (periodic table) , abelian group , pure mathematics , finite field , field (mathematics) , finite group , property (philosophy) , arithmetic , galois theory , cohomology , galois group , cyclic group , discrete mathematics , chemistry , organic chemistry , philosophy , epistemology
Let G be a finite group and let K be a hilbertian field. Many finite groups have been shown to satisfy the arithmetic lifting property over K , that is, every G ‐Galois extension of K arises as a specialization of a geometric branched covering of the projective line defined over K . The paper explores the situation when a semidirect product of two groups has this property. In particular, it shows that if H is a group that satisfies the arithmetic lifting property over K and A is a finite cyclic group then G = A ⋊ H also satisfies the arithmetic lifting property assuming the orders of H and A are relatively prime and the characteristic of K does not divide the order of A . In this case, an arithmetic lifting for any A ≀ H ‐Galois extension of K is explicitly constructed and the existence of the arithmetic lifting for any G ‐Galois extension is deduced. It is also shown that if A is any abelian group, and H is the group with the arithmetic lifting property then A ≀ H satisfies the property as well, with some assumptions on the ground field K . In the construction properties of Hilbert sets in hilbertian fields and spectral sequences in étale cohomology are used.