If I have the exact sequence
is the projective limit indicated with the obvious projection maps, then do I know that
is isomorphic to
is the canonical projection map to
which is the one furnished by the definition of
as an inverse limit to the initial bit
I feel like the answer must be something like "yes, because the fiber of a discrete group is just that group crossed with the fiber", but I've never done this before, and I don't see why the set structure which I definitely believe is a set of cardinality m crossed with
should be inherited in the group structure.
Really I'm looking for just a bit of elaboration on how the topology and the algebra are interacting here and what gives the result if it is true, and why it is false if it is false, it doesn't seem hard, but I need to know why it works if I'm going to understand it or use the idea in the future. Thanks, and any help appreciated.
Well, I trust you have seen the fact that the p-adic solenoid
has a unique cyclic subgroup of order m, for every
prime to p. That will give you a splitting
if you check the construction (in
the nonidentity elements of
don't live in
On the other hand, when
you cannot have such a splitting --- the p-adic solenoid has no
-torsion (the composition
must be trivial).