15.2. Subgroups

Try these first

1. 1.

   Define a subgroup in your own words.

   Hint

   Think of it as a smaller group that still behaves like the big group.

2. 2.

   What are the properties needed for a subset to be a subgroup?

   Hint

   There are three main points to remember.

3. 3.

   Is a non-empty subset of a group always a subgroup?

   • True
   • False

   Hint

   Think about the defining properties of subgroups.

4. 4.

   What must be true for any two elements in a subgroup?

   • Their sum must be in the subgroup
   • One of them must be the identity
   • Their product must be zero

   Hint

   Remember the definition of closure!

5. 5.

   Given the group Z_12, identify all possible subgroups and their generators.

   Hint

   Consider how elements combine under addition mod 12.

6. 6.

   Prove that every subgroup of a cyclic group is cyclic.

   Hint

   Reflect on what it means for a generator to create elements of the group.