#
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 $G$ 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 $G$*.

## 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 $H \hookrightarrow G$ 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

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

## Examples

Revised on April 9, 2014 10:27:39
by

Urs Schreiber
(77.251.114.72)