This is an exercise in the book "A book of abstract algebra" by Pinter, 3E.
Let's devise a coin game.
Imagine there are two coins, each can be placed in either location $A$ or $B$ . Moreover, each coin can be flipped.
Define 8 moves:
$M_1$ - flip over the coin at $A$
$M_2$ - flip over the coin at $B$
$M_3$ - flip over both coins
$M_4$ - switch the coins
$M_5$ - flip the coin at A, then switch
$M_6$ - flip the coin at B, then switch
$M_7$ - flip both coins, then switch
$I$ - do not change anything
Example: $M_4 * M_1 = M_2 * M_4 = M_6$
"switch" means "change places"
I will spare you from what the exercise is asking to do. But I have noticed something playing with the problem. One of the aspects of the definitions of an operation implies that if $*$ is an operation, then $a*b$ is unambiguously and uniquely defined. An operation on the above set $G = \{ M_1, M_2, M_3, M_4, M_5, M_6, M_7, I \}$ has been defined as "performing any two moves in succession". Imagine now that we have done: $M_1 * M_2$ , well then the result of this operation can be expressed with two elements in the set, namely: $M_1 * M_2 = M_3 = M_7$ and so this creates ambiguity as to what this specific operation ("performing any two moves in succession") assigns two members of the set ( $M_1, M_2$ ) to. As such, can we even consider $\langle G, *\rangle$ a group (I obviously must be missing / misunderstanding something)?