1. Chapter 3: Finite Groups; Subgroups  Terminology and Notation  Subgroup Tests  Examples of Subgroups

2. e Examples:

4. 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 have a, 2a, 3a, 4a, …… is never 0, hence |a|=∞

5. 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 write H<G and we say H is a proper subdroup of G.

6. 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.

7. Subgroup Tests

9. Examples 1

10. Example 2

12. Example 3

16. Examples

17. Examples\ continue

20. Definition: Center of a Group

21. hjhjh Proof: By the 2-steps test The center of a group G is a subgroup of G. The center of a group G is a subgroup of G.

23. Definition: Centralizer of a

25. Proof: Exercise 25