Originally Posted by

**VonNemo19** Hi.

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

$\displaystyle M_1:$ flip the coin at A.

$\displaystyle M_2:$ flip the coin at B.

$\displaystyle M_3:$ flip both coins.

$\displaystyle M_4:$ switch the coins.

$\displaystyle M_5:$ flip the coin at A, then switch.

$\displaystyle M_6:$ flip the coin at B, then switch.

$\displaystyle M_7:$ flip both, then switch.

$\displaystyle I:$ do nothing.

We may consider an operation on the set $\displaystyle \{I,M_1,...,M_7\}$, 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 $\displaystyle (G,*)$.

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,

$\displaystyle M_1*M_2=?$.

Can anyone help me see this?

*edit>

OK. So, what about $\displaystyle M_5*M_2$ ? 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 $\displaystyle M_5$ 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 $\displaystyle M_2$ and 1 now shows at B. This is where we started. So, is $\displaystyle M_5*M_2=I$ or $\displaystyle M_5*M_2=M_4$. **To me it seems as if $\displaystyle *$ is not uniquely defined.**