Hi.

Imagine 2 coins on a table, at positions A and B. In the game there are 8 possible moves:

flip the coin at A.

flip the coin at B.

flip both coins.

switch the coins.

flip the coin at A, then switch.

flip the coin at B, then switch.

flip both, then switch.

do nothing.

We may consider an operation on the set

, which consists of performing any 2 moves in succession. EG:

If we switch coins, then flip at A, this is the same as first flipping over the coin at B then switching.
Write the table for

.

Ok, so I understand the notion of group, I understand how to create operation tables, but I'm having trouble visualizing the equivalences between a pair of elements and the associated element which is the pair's product. For example,

.

Can anyone help me see this?

*edit>

OK. So, what about

? The way that I think about this is like, if we let the coins have sides 1 and 2 at points A and B, respectively, then if side 1 is showing at both A and B, and

is made, then 1 becomes 2 at A and then moves to B. So, we have 1 at A and 2 at B. Now, we do

and 1 now shows at B. This is where we started. So, is

or

.

**To me it seems as if is not uniquely defined.**