nLab
subgroup
Context
Group Theory
group theory
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Contents
Idea
A subgroup of a group is a “smaller” group sitting inside .
Definition
A subgroup is a subobject in the category Grp of groups: a monomorphism of groups
Here is a subgroup of .
Special cases
Properties
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
Examples
Last revised on July 14, 2016 at 05:55:27.
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