Represent each element of the group \(G = \Z_{12}\) with a poker chip. Separate out the subgroup \(H = \langle 4 \rangle = \{0, 4, 8\}\text{.}\) Partition the poker chips into the cosets of \(\langle 4 \rangle\) in \(\Z_{12}\text{.}\) Put each coset into its own paper bag.
If the bags are labeled \(A\text{,}\)\(B\text{,}\)\(C\text{,}\) and \(D\text{,}\) you might wonder if we can define an operation on the bags. What should \(A + B\) be? What about \(B+D\text{?}\) How can you use the chips in the bag to help you define the operation?
Will this operation you define be associative? Will there be an identity (which bag is it)? Will every bag have an inverse bag? Create the Cayley table for the bag-group.
Let’s switch groups. Use chips to represent the elements of \(S_3\text{.}\) One subgroup is \(H = \{(1), (12)\}\text{.}\) Form cosets/bags. Now try to define the operation on the bags. What is \(BC\text{?}\) Are you sure? (We use multiplication instead of addition, to match the way we write the operation on \(S_3\text{.}\))
What if \(H = \{(1), (123), (132)\}\text{?}\) Now there are only 2 bags. Will the operation on the bags, induced by the chips, be well-defined? What does that question even mean.
