Group theory MOC

Subgroup

A subgroup is a subset of a group such that is a group under the same operations #m/def/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 . #m/thm/group

Two step subgroup test

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

Finite subgroup test

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

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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