nLab
subgroup

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

$$K\hookrightarrow G.$$K \hookrightarrow G \,.

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

Special cases

• normal subgroup

• stabilizer subgroup

• maximal compact subgroup

• torsion subgroup

• Borel subgroup

• parabolic subgroup

• generalized Fitting subgroup

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 $BH\to BG$ 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

$$G/H\to BH\to BG.$$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

$$\begin{array}{ccccc}K/H& \to & G/H& \to & G/K\\ \downarrow & & \downarrow & & \downarrow \\ BH& \to & BH& \to & BK\\ \downarrow & & \downarrow & & \downarrow \\ BK& \to & BG& \to & BG\end{array}$$\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 }

Related concepts

• Klein geometry, Cartan geometry

• index of a subgroup