Pro-C groups
A pro-C group is built from finite groups in a chosen class C by taking inverse limits. This library develops ways to study such groups through their finite quotients. The universal properties of completions and free groups explain how to construct continuous homomorphisms, while the open-subgroup theorem computes the ranks of open subgroups in the finite-rank free case.
The library also studies how defining relations constrain the structure of a group. Fox derivatives and Crowell sequences express relations in modules, and a weighted Golod–Shafarevich criterion detects infinite pro-p groups. For profinite abelian groups, continuous characters recover the original group; Tate modules describe how compatible torsion points fit together. The individual results specify the hypotheses on C and the groups involved.
Selected theorems and definitions
- definition
What a pro-C completion means
Completion is characterized by a dense map through which continuous maps to pro-C groups factor uniquely.
- definition
Free pro-C groups
A continuous choice of generator images extends uniquely to a continuous homomorphism.
- theorem
The Schreier rank of an open subgroup
An open subgroup of a finite-rank free pro-C group is free, with the expected index-dependent rank.
- theorem
The relative Fox fundamental formula
A group-ring boundary decomposes into Fox derivatives times the boundaries of the generators.
- theorem
The discrete Crowell exact sequence
A surjective group homomorphism yields an exact sequence connecting its kernel to its derived module.
- theorem
The separated completed Crowell sequence
The kernel–differential–augmentation sequence is exact in the stated finite-rank pro-C setting.
- theorem
A weighted Golod–Shafarevich infinitude criterion
A nonpositive presentation polynomial certifies that an actual pro-p group is infinite.
- theorem
Pontryagin biduality for profinite abelian groups
Evaluation identifies a profinite abelian group with its continuous character bidual.
- definition
The full Tate module of an abelian group
Compatible torsion points at every positive integer form a torsion-free inverse-limit object.
- theorem
Recognizing the profinite integers
A topological generator and finite quotients of every order characterize the additive group of profinite integers.
References used in these guides
- Luis Ribes and Pavel Zalesskii. Profinite Groups. Springer-Verlag Berlin Heidelberg, 2010. Second edition; Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, volume 40.
- Ralph H. Fox. Free Differential Calculus. I: Derivation in the Free Group Ring. Annals of Mathematics, 57 (3), 547–560, 1953.
- R. H. Crowell. The Derived Module of a Homomorphism. Advances in Mathematics, 6, 210–238, 1971. Original article; the linked PDF reproduces the published text.
- E. S. Golod and I. R. Shafarevich. On the class field tower. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 28 (2), 261–272, 1964. Original article in Russian.
- L. Pontrjagin. The Theory of Topological Commutative Groups. Annals of Mathematics, 35 (2), 361–388, 1934.
- J. T. Tate. p-Divisible Groups. Springer, Berlin, Heidelberg, 158–183, 1967. In Proceedings of a Conference on Local Fields.