15.2.5. Cosets

Try these first

1. 1.

   Define a left coset in your own words.

   Hint

   Consider which side the group element appears.

2. 2.

   What is the significance of Lagrange's theorem?

   Hint

   Think about relationships between subgroups and their parent groups.

3. 3.

   What is a left coset?

   • A set of elements formed by subtracting a subgroup from a group
   • A collection formed by multiplying a group element with a subgroup
   • The same as a right coset

   Hint

   Think about the multiplication order.

4. 4.

   True or False: The order of a subgroup divides the order of the group.

   • True
   • False

   Hint

   Consider the implications of subgroup sizes.

5. 5.

   Given a group of integers under addition modulo 8, identify the subgroup generated by 2. List the left cosets of this subgroup.

   Hint

   Calculate by adding the group element to each member of the subgroup.

6. 6.

   In a group of order 12, if a subgroup has order 3, determine the number of distinct left cosets.

   Hint

   Use Lagrange's theorem to divide the total group order by the order of the subgroup.