nLab subgroup

Idea

A subgroup of a group is a subset equipped with the “same” group operation. As-in, a subobject in the category Grp of groups; hence a monomorphism of groups

$$K \hookrightarrow G \,.$$

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

Definition

A subset $H$ of the set $G$ underlying a group is a subgroup of $G$ iff the restriction of the group operation • to $H$ is a group. For all subgroups $H$ of a group $G$,

• the identity element $e\in G$ is in $H$,

• for all $h\in H$, the inverse $h^-^1\in G$ is in $H$,

• for non-negative integers $n$ and $m$ $H$={$g ^n h ^-^m$|$g \in H \wedge h \in H$}.

For $A$ a subset of the automorphisms group $Aut(G)$ of a group $G$, a subgroup $H \subseteq G$ is called an $A$-invariant subgroup of $G$ if for $h \in H$ and $\alpha \in A$ then $\alpha (h) \in H$. A subgroup $H$ of a group $G$ is called:

• a characteristic subgroup of $G$ if $H$ is $Aut(G)$-invariant.

• a normal or self-conjugate subgroup of $G$ is $H$ is $Inn Aut(G)$-invariant. As-in, if $g \in G$ and $h \in H$ then $g ^- ^1hg \in H$.

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 $\mathbf{B}H \to \mathbf{B}G$ 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

• centralizer subgroup

• normalizer subgroup

• stabilizer subgroup

• conjugate subgroup

• congruence subgroup

• Young subgroup

Related concepts

• lattice of subgroups

• maximal subgroup

• index of a subgroup

• Klein geometry, Cartan geometry

• BN-pair

• subring, submodule

References

Discussion in univalent foundations of mathematics (homotopy type theory with the univalence axiom, but for 1-groups):

• Martín Escardó, Subgroups, §33.12 in: Introduction to Univalent Foundations of Mathematics with Agda [arXiv:1911.00580, webpage]

  (in Agda)

• Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: Chapter 5 of: Symmetry (2021) $[$pdf$]$

See also

• Wikipedia, Subgroup