Chapter 3: Finite Groups; Subgroups

Terminology and Notation

Subgroup Tests

Examples of Subgroups

e

Examples:

Examples: Example 1 :The group U(15)={1,2,4,7,8,11,13,14}

Example 2: The group

Example 3: The group (Z,+) Note that for non zero element a, we havea, 2a, 3a, 4a, …… is never 0, hence |a|=∞

10Z

Definition: Subgroup

If H is a subgroup of G, we write H≤ G If H is a subgroup of G and H is a proper subset, we writeH<G and we say H is a proper subdroup of G.

Examples {e} is a subgroup of any group G called the trivial subgroup.

Any subgroup other than {e} is called non trivial.

Any group G is a subgroup of itself.

Subgroup Tests

Examples 1

Example 2

Example 3

Examples

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Definition: Center of a Group

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Proof: By the 2-steps test

The center of a group G is a subgroup of G.

Definition: Centralizer of a

Proof: Exercise 25