Let's assume that the cost of each ticket is "x".
For Group 1, the cost of 9 tickets without the discount would be 9x. Since they received a $120 discount, the total amount they spent on tickets is 9x - $120.
For Group 2, the cost of 3 tickets without the discount would be 3x. Since they received a $30 discount, the total amount they spent on tickets is 3x - $30.
Since both groups spent the same total amount of money on tickets, we can set up the following equation:
9x - $120 = 3x - $30
Simplifying the equation, we get:
6x = $90
Dividing both sides by 6, we get:
x = $15
Therefore, the cost of each ticket is $15. Answer: $\boxed{15}$.