nLab
subgroup

Contents

Idea

A subgroup of a group GG is a “smaller” group KK sitting inside GG.

Definition

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

KG. K \hookrightarrow G \,.

Here KK is a subgroup of GG.

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 HGH \hookrightarrow G a sub-Lie group inclusion write BHBG\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/HG/H: there is a homotopy fiber sequence

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

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

K/H G/H G/K BH BH BK BK BG BG \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)