15.2.1. Definition of Subgroups

Try these first

1. 1.

   What is a subgroup?

   Hint

   Think about the definition of a group you learned before.

2. 2.

   List the three main properties a subset must satisfy to be a subgroup.

   Hint

   Recall our classroom discussion on subgroup conditions.

3. 3.

   What property must a subset satisfy to be a subgroup?

   • Contains the identity
   • Closure under operation
   • All of the above

   Hint

   Think of the subgroup conditions we discussed.

4. 4.

   True or False: A subgroup can be empty.

   • True
   • False

   Hint

   Remember the definition of a group.

5. 5.

   Given a group G of order 15, list the possible orders of its subgroups. Prove your reasoning using Lagrange’s theorem.

   Hint

   Find factors of 15.

6. 6.

   Construct a subgroup for the group of permutations on three objects. Show all properties hold.

   Hint

   Consider how permutations combine and find pieces that remain closed.