nLab
subgroup

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

KG.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 HG a sub-Lie group inclusion write BHBG 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/HBHBG.G/H \to \mathbf{B}H \to \mathbf{B}G \,.

Now let HKG 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 }

Revised on January 24, 2013 17:00:51 by Urs Schreiber (82.113.99.233)