15.2.4. Generating Cyclic Subgroups

Try these first

1. 1.

   Define what makes a subgroup.

   Hint

   Recall the group axioms.

2. 2.

   What is the closure property?

   Hint

   Think of operations in a group.

3. 3.

   What is the definition of a subgroup?

   • A subgroup can only be empty
   • A non-empty subset that is a group under the same operation
   • Any subset of a group

   Hint

   Consider the properties related to subsets.

4. 4.

   Can closure alone guarantee a subset is a subgroup in infinite groups?

   • True
   • False

   Hint

   Think about the implications of it being infinite.

5. 5.

   Prove that every cyclic subgroup is abelian.

   Hint

   Think of how elements generated by powers commute.

6. 6.

   Given the group of real numbers under addition, show that subsets like {x ∈ ℝ | x ≤ 0} and {x ∈ ℝ | x ≥ 0} are not subgroups.

   Hint

   Can you find elements in the subsets that fail closure?