Group theory MOC

Subgroup

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

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

Tests for subgroups

Let be a group and be a inhabited subset. Additionally we define predicate so that . The following hold:¹

One step subgroup test

Theorem. Iff whenever , then is a subgroup of . group

Proof

Since is inhabited there exists , then with clearly , must be closed under inversion, since letting for any we have . Now we can show that is closed in general: For any we have and therefore .

Application

1. Show
2. Assume and
3. Prove

Two step subgroup test

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

Proof

Since is inhabited there exists in , thus and thus . Thence is a subgroup of .

Application

Much the same as above, but with

3. Prove and .

Finite subgroup test

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

Proof

Take any . Since is closed we may construct a sequence . Since is finite, by the Pigeonhole principle the sequence must have repeated elements, so that for some we have . Then and hence , so . Therefore is closed under the inverse and the binary operation, and is thus a subgroup of by the Two step subgroup test.

Examples of subgroups

• For any element we can generate a Cyclic subgroup .
• 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.

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Footnotes

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