subgroup (changes) in nLab

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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

K↪G. K \hookrightarrow G \,.

Here KK is a subgroup of GG.

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↪GH \hookrightarrow G a sub-Lie group inclusion write BH→BG\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/H→BH→BG. G/H \to \mathbf{B}H \to \mathbf{B}G \,.

Now let H↪K↪GH \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

• centralizer subgroup

• normalizer subgroup

• stabilizer subgroup

• conjugate subgroup

• congruence subgroup

• Young subgroup

• 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 , but with mostly the for 1-groups):univalence axiom, but for 1-groups):

• Marc Bezem ,

  Martín Escardó, Subgroups, §33.12 in: Introduction to Univalent Foundations of Mathematics with Agda [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580), webpage]

  Ulrik Buchholtz ,

  (in Agda)

  Pierre Cagne , Bjørn Ian Dundas, Daniel R. Grayson: Chapter 5 of: Symmetry (2021) [[pdf]]

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

See also

• Wikipedia, Subgroup