Difference between revisions of "P Adic Solenoid"
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The canonical group homomorphisms | The canonical group homomorphisms | ||
<center><cmath>\varphi_n:\mathbb{R}/p^{n+1}\mathbb{Z}\to \mathbb{R}/p^n\mathbb{Z},~x\pmod{p^{n+1}\mathbb{Z}}\mapsto x\pmod{p^n\mathbb{Z}}</cmath></center> | <center><cmath>\varphi_n:\mathbb{R}/p^{n+1}\mathbb{Z}\to \mathbb{R}/p^n\mathbb{Z},~x\pmod{p^{n+1}\mathbb{Z}}\mapsto x\pmod{p^n\mathbb{Z}}</cmath></center> | ||
| − | form a projective system <math>(\mathbb{R}/p^n\mathbb{Z},\varphi_n)_{n\ge 0}</math> of topological groups. The '''<math>p</math> adic solenoid <math>\mathbb{S}_p</math>''' is a projective limit <math>\mathbb{S}_p=\lim_{\longleftarrow}\mathbb{R}/p^n\mathbb{Z}</math> of the projective system <math>(\mathbb{R}/p^n\mathbb{Z},\varphi_n)_{n\ge 0}</math>. | + | form a projective system <math>(\mathbb{R}/p^n\mathbb{Z},\varphi_n)_{n\ge 0}</math> of topological groups. The '''<math>p</math> adic solenoid <math>\mathbb{S}_p</math>''' is a projective limit <math>\mathbb{S}_p=\lim_{\longleftarrow}\mathbb{R}/p^n\mathbb{Z}</math> of the projective system <math>(\mathbb{R}/p^n\mathbb{Z},\varphi_n)_{n\ge 0}</math>. It is both compact and connected. |
Latest revision as of 14:05, 2 July 2022
The canonical group homomorphisms
![\[\varphi_n:\mathbb{R}/p^{n+1}\mathbb{Z}\to \mathbb{R}/p^n\mathbb{Z},~x\pmod{p^{n+1}\mathbb{Z}}\mapsto x\pmod{p^n\mathbb{Z}}\]](http://latex.artofproblemsolving.com/d/6/3/d63aef7196b69bcdc402968fc9be3f2a518ab6b0.png)
form a projective system 
of topological groups. The 
adic solenoid 
is a projective limit 
of the projective system . It is both compact and connected.
