nLab subgroup

Contents

Idea

A subgroup of a group $G$ is a “smaller” group $K$ sitting inside $G$ .

A subgroup is a subobject in the category Grp of groups: a monomorphism of groups

K \hookrightarrow G \,.

Here $K$ is a subgroup of $G$ .

Every subgroup of a free group is itself free. This is the statement of the Nielsen-Schreier theorem.

For $H \hookrightarrow G$ a sub-Lie group inclusion write $\mathbf{B}H \to \mathbf{B}G$ for the induced map on delooping Lie groupoids. The homotopy fiber of this map (in Smooth∞Grpd) is the coset space $G/H$ : there is a homotopy fiber sequence

G/H \to \mathbf{B}H \to \mathbf{B}G \,.

Now let $H \hookrightarrow K \hookrightarrow G$ be a sequence of two subgroup inclusions. By the above this yields the diagram

\array{ K/H &\to& G/H &\to& G/K \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}H &\to& \mathbf{B}H &\to& \mathbf{B}K \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}K &\to& \mathbf{B}G &\to& \mathbf{B}G }