15.2.3. Corollary on Finite Groups

Try these first

1. 1.

   What is a subgroup?

   Hint

   Think about what group properties must hold.

2. 2.

   Define the closure property.

   Hint

   Consider how operations behave in a group.

3. 3.

   What is a subgroup?

   • A subset that meets certain criteria
   • Any subset of a group
   • A group with fewer elements

   Hint

   Consider the definition of groups.

4. 4.

   True or False: Every subgroup has the same number of elements as its parent group.

   Hint

   Think about examples with smaller subsets.

5. 5.

   Given the set of real numbers, prove or disprove that the set of negative rational numbers is a subgroup under addition.

   Hint

   Check closure when adding two elements from the subset.

6. 6.

   Consider a group of order 12 and a subgroup of order 3. How many distinct cosets can be formed from this subgroup?

   Hint

   Utilize the relationship from Lagrange’s theorem.