Group theory MOC

Subgroup

A subgroup is a subset of a group 𝐻 ⊆𝐺 such that 𝐻 is a group under the same operations group , i.e. 𝐻 is

• closed under the group operation
• contains the inverse of every element

Tests for subgroups

Let 𝐺 be a group and 𝐻 ⊆𝐺 be a inhabited subset. Additionally we define predicate 𝑝 so that 𝐻 ={𝑥 ∈𝐺 ∣𝑝(𝑥)}. The following hold:¹

One step subgroup test

Theorem. Iff 𝑎𝑏−1 ∈𝐻 whenever 𝑎,𝑏 ∈𝐻, then 𝐻 is a subgroup of 𝐺. group

Proof

Since 𝐻 is inhabited there exists 𝑥 ∈𝐻, then with 𝑎 =𝑏 =𝑥 clearly 𝑒 ∈𝐻, 𝐻 must be closed under inversion, since letting 𝑎 =𝑒 for any 𝑏 ∈𝐻 we have 𝑒𝑏−1 =𝑏−1 ∈𝐻. Now we can show that 𝐻 is closed in general: For any 𝑎,𝑏 ∈𝐻 we have 𝑏−1 ∈𝐻 and therefore 𝑎𝑏 =𝑎(𝑏−1)−1 ∈𝐻.

Application

1. Show 𝑝(𝑒)
2. Assume 𝑝(𝑎) and 𝑝(𝑏)
3. Prove 𝑝(𝑎𝑏−1)

Two step subgroup test

Theorem. Iff 𝐻 is closed under the binary operation and under the inverse, then 𝐻 is a subgroup of 𝐺. group

Proof

Since 𝐻 is inhabited there exists 𝑥 in 𝐻, thus 𝑥−1 ∈𝐻 and thus 𝑒 =𝑥𝑥−1 ∈𝐻. Thence 𝐻 is a subgroup of 𝐺.

Application

Much the same as above, but with

3. Prove 𝑝(𝑎−1) and 𝑝(𝑎𝑏).

Finite subgroup test

Theorem. 𝐻 is finite and closed under the binary operation, then it is a subgroup of 𝐺. group

Proof

Take any 𝑥 ∈𝐻. Since 𝐻 is closed we may construct a sequence (𝑥𝑛)∞𝑛=1 ∈𝐻. Since 𝐻 is finite, by the Pigeonhole principle the sequence must have repeated elements, so that for some 1 <𝑖 <𝑗 we have 𝑥𝑖 =𝑥𝑗. Then 𝑎𝑖−𝑗 =𝑒 and hence 𝑎𝑎𝑖−𝑗−1 =𝑒, so 𝑎−1 =𝑎𝑖−𝑗−1 ∈𝐻. Therefore 𝐻 is closed under the inverse and the binary operation, and is thus a subgroup of 𝐺 by the Two step subgroup test.

Examples of subgroups

• For any element 𝑎 ∈𝐺 we can generate a Cyclic subgroup ⟨𝑎⟩ ={…,𝑎−2,𝑎−1,𝑒,𝑎,𝑎2,…}.
• The Centre of a group is a subgroup containing elements that commute with all elements.
• Similarly the Centralizer in a group is a subgroup containing elements that commute with a given element.
• Torsion subgroup of an abelian group
• The order of a subgroup divides the order of a group

Properties

• The intersection of subgroups is a subgroup
• Subgroups may be Conjugate subgroups if they can be derived from each other by conjugation. A subgroup with no other conjugate subgroups is called a Normal subgroup.

---

tidy | en | SemBr

Footnotes

1. 2017, Contemporary Abstract Algebra, pp. 62–64 ↩