Cohomology and Extensions
A subgroup of a group is a “smaller” group sitting inside .
A subgroup is a subobject in the category Grp of groups: a monomorphism of groups
Here is a subgroup of .
Of free groups
Every subgroup of a free group is itself free. This is the statement of the Nielsen-Schreier theorem.
Of Lie groups
For a sub-Lie group inclusion write for the induced map on delooping Lie groupoids. The homotopy fiber of this map (in Smooth∞Grpd) is the coset space : there is a homotopy fiber sequence
Now let be a sequence of two subgroup inclusions. By the above this yields the diagram
Revised on November 1, 2013 06:42:04
by Urs Schreiber