15.2.6. Lagrange's Theorem

Try these first

1. 1.

   Define a subgroup in your own words.

   Hint

   Think about the properties a subset must hold to be a group.

2. 2.

   What does Lagrange's Theorem state?

   Hint

   Consider how the number of elements correlates.

3. 3.

   What does Lagrange's Theorem state?

   • The sum of group elements
   • The order of a subgroup divides the order of the group
   • Subgroups are always finite

   Hint

   Recall the relationship between group sizes.

4. 4.

   True or False: A subgroup can be empty.

   • True
   • False

   Hint

   Link to the definition of subgroup.

5. 5.

   Given the group S3 (symmetric group of order 6), identify the subgroups and demonstrate how Lagrange’s Theorem applies.

   Hint

   Use the concepts of permutations to explore subgroup structures.

6. 6.

   Construct a non-trivial group of order 12 and apply Lagrange’s Theorem to determine possible subgroup orders.

   Hint

   List divisors of the group order and check group structure properties.