Mathematics.

groups

Subgroups

Abstract Algebra I35 minDifficulty6 out of 10

You should know: group mathematics

Overview

A subgroup is a subset of a group that is itself a group under the same operation. Subgroups let us study a large group by decomposing it into smaller, self-contained pieces, and they are the objects that Lagrange's theorem and the classification of group structure revolve around.

Intuition

If a group is a full set of 'moves' that can be combined and undone, a subgroup is a smaller collection of moves that never needs to leave its own list: combining any two of its moves, or undoing one, always produces another move already in the list. The rotations of a square (ignoring the flips) form a subgroup of the square's full symmetry group, because rotating twice still gives a rotation.

Formal Definition

Definition

Let (G, ·) be a group. A subset H ⊆ G is a subgroup of G, written H ≤ G, if H is itself a group under the operation inherited from G. Equivalently, by the subgroup test:

H≠∅H \neq \emptyset
Nonempty (usually shown via e \in H)
∀a,b∈H, a⋅b∈H\forall a, b \in H,\ a \cdot b \in H
Closure under the operation
∀a∈H, a−1∈H\forall a \in H,\ a^{-1} \in H
Closure under inverses

Notation

NotationMeaning
H≤GH \leq GH is a subgroup of G
H<GH < GH is a proper subgroup of G (H ≠ G)
⟨a⟩\langle a \rangleThe subgroup generated by element a
[G:H][G:H]Index of H in G — number of distinct cosets of H

Properties

Trivial subgroup

{e}≤G and G≤G are always subgroups.\{e\} \leq G \text{ and } G \leq G \text{ are always subgroups.}

One-step subgroup test

H is a subgroup iff H≠∅ and ab−1∈H ∀a,b∈H.H \text{ is a subgroup iff } H \neq \emptyset \text{ and } ab^{-1} \in H \ \forall a,b \in H.

Lagrange's theorem

If G is finite and H≤G, then ∣H∣ divides ∣G∣.\text{If } G \text{ is finite and } H \leq G, \text{ then } |H| \text{ divides } |G|.

Condition: G finite

Intersection of subgroups

If H,K≤G then H∩K≤G.\text{If } H, K \leq G \text{ then } H \cap K \leq G.

Example: The intersection of two subgroups is always a subgroup, but the union generally is not.

Applications

Subgroup structure underlies the security proofs of Diffie-Hellman key exchange, where the relevant subgroup of a multiplicative group modulo p determines cryptographic strength.

Worked Examples

  1. Nonempty: 0 ∈ 2ℤ since 0 is even.

    0∈2Z0 \in 2\mathbb{Z}

  2. Closure: the sum of two even integers is even.

    2m+2n=2(m+n)∈2Z2m + 2n = 2(m+n) \in 2\mathbb{Z}

  3. Inverses: the negative of an even integer is even.

    −(2m)=2(−m)∈2Z-(2m) = 2(-m) \in 2\mathbb{Z}

Answer: 2ℤ satisfies the subgroup test, so it is a subgroup of (ℤ, +).

Practice Problems

Difficulty 5/10

Explain why {0, 3} is NOT a subgroup of (ℤ₆, +) even though it's a subset containing the identity.

Difficulty 4/10

A finite group has order 12. Which of the following CANNOT be the order of one of its subgroups?

Common Mistakes

Common Mistake

Believing the union of two subgroups is always a subgroup.

The union H ∪ K is a subgroup only in special cases (e.g. one contains the other); in general it fails closure. The intersection H ∩ K, however, is always a subgroup.

Common Mistake

Forgetting to check that the identity is actually shared with the parent group.

A subgroup must use the identity of G, not a new one — verify e_G ∈ H rather than just checking H has 'an' identity.

Quiz

By Lagrange's theorem, the order (size) of a subgroup H of a finite group G must:
To confirm a nonempty subset H of a group is a subgroup, it suffices to check:

Summary

  • A subgroup H ≤ G is a subset that forms a group under G's inherited operation.
  • The one-step test: H is nonempty and closed under a·b⁻¹ for all a, b ∈ H.
  • Lagrange's theorem: for finite G, the order of any subgroup divides |G|.
  • Intersections of subgroups are subgroups; unions generally are not.
  • Subgroups are the building blocks used to analyze cosets, quotient groups, and homomorphism images.

References

  1. WebsiteWikipedia — Subgroup
  2. BookDummit, D. & Foote, R. Abstract Algebra, 3rd ed., Ch. 2.