Contents

group theory

# Contents

## Idea

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

## Definition

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

$K \hookrightarrow G \,.$

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

## 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 $H \hookrightarrow G$ 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

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

## References

Discussion in univalent foundations of mathematics (homotopy type theory with the univalence axiom, but for 1-groups):