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

Related concepts

• Klein geometry, Cartan geometry