###### ObjectivesLearning Goals for this Chapter

- Determine the order of a given group, and distinguish between finite and infinite groups.
- Determine the order of given elements in a group, and distinguish between elements of finite order and infinite order.
- Rewrite expressions in a finite group using positive exponents only.
- Determine whether a given subset of a group is a subgroup, using the definition and/or selecting and applying a subgroup test.
- Determine the center of a given group, and the centralizer of a given element in a group.
- Discover and prove relationships among the concepts of center, centralizer, and abelian groups.